Central European Journal of Mathematics

CEJM 3(4) 2005 580–590

Limit theorems for the Estermann zeta-function. II∗ Antanas Laurinˇcikas† Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

Received 11 May 2005; accepted 6 July 2005 Abstract: A limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for the Estermann zeta-function is obtained. c Central European Science Journals. All rights reserved.

Keywords: Estermann zeta-function, distribution, probability measure, random element, space of analytic functions, space of meromorphic functions, weak convergence MSC (2000): 11M41

1

Introduction

The paper is devoted to application of the statistical approach in value distribution of the Estermann zeta - function. This approach has been proposed by H. Bohr in the third decade of the last century and realized by Bohr and Jessen [2-3], in the investigation of the Riemann zeta - function. Later, many mathematicians developed the Bohr - Jessen theory, see, for example, [6]. The Estermann zeta-function E(s; kl , α), s = σ + it, with parameters kl , (k, l) = 1, and α is defined, for σ > max(1, 1 + Reα), by   ∞ X k σα (m) k E(s; , α) = exp 2πim , s l m l m=1 P where σα (m) = d|m dα is the generalized divisor function. The function E(s; kl , α) is meromorphically continuable to the whole complex plane with two simple poles s = 1 and s = 1 + α if α 6= 0, and one double pole s = 1 if α = 0. ∗ †

Partially supported by Lithuanian Foundation of Studies and Science E-mail: [email protected]

A. Laurinˇcikas / Central European Journal of Mathematics 3(4) 2005 580–590

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The Estermann zeta - function has not the Euler product, therefore its properties significantly differ, for example, from those of the Riemann zeta - function. However, a modified approach allows one to characterize the statistical behavior of E(s; kl , α) by probabilistic limit theorems. In [8] a limit theorem in the sense of the weak convergence of probability measures on the complex plane C for the function E(s; kl , α) has been obtained. To state that theorem we need some notation. Denote by meas {A} the Lebesgue measure of the set A ⊂ R, and let, for T > 0, νTt (...) = T1 meas {t ∈ [0, T ] : ...} , where in place of dots a condition satisfied by t is to be written. Let B(S) stand for the class of Borel sets of the space S. Then in [8] it was proved that, for σ > 12 , the probability measure
νTt E(σ + it; kl , α) ∈ A , A ∈ B(C), converges weakly to the distribution of the random element   ∞ X k σα (m)ω(m) exp 2πim mσ l m=1 as T → ∞. The later random element is defined on the probability space (Ω, B(Ω), mH ), Q where Ω = γp , and mH denotes the probability Haar measure on (Ω, B(Ω)). Moreover, p Q r γp , for all primes p, is the unit circle on the complex plane, ω(m) = ω (p), where pr ||m

pr || m means that pr | m but pr+1 ∤ m, and ω(p) denotes the projection of ω ∈ Ω onto the coordinate space γp . The aim of this paper is to obtain a limit theorem in the space of meromorphic S functions for the function E(s; kl , α). Let C∞ = C {∞} be the Riemann sphere, and let d denote the spheric metric defined by the formulae d(s1 , s2 ) = p

2|s1 − s2 | p , 1 + |s1 |2 1 + |s2 |2

2 d(s, ∞) = p , 1 + |s|2

d(∞, ∞) = 0,

s, s1 , s2 ∈ C. Denote by M(G) the space of meromorphic on G functions f : G → (C∞ , d) equipped with the topology of uniform convergence on compacta. In this topology, a sequence {fn } ⊂ M(G) converges to f ∈ M(G) if d(fn (s), f (s)) → 0 as n → ∞ uniformly on compact subsets of G. The space H(G) of analytic on G functions is a subspace of M(G). In [8] it was noted that E(s; kl , α) = E(s − α; kl , −α). Therefore, without loss of generality we can suppose that Reα ≤ 0. For such α, σα (m) ≤ d(m), where d(m) is the divisor function. Hence, in view of the well - known mean value estimate for d(m), X X σα2 (m) ≤ d2 (m) ≪ n log3 n. m≤n

m≤n

This, the pairwise orthogonality of the random variables ω(m), and Rademacher’s theorem [9] show that the series   ∞ X k σα (m)ω(m) k E(s; , α, ω) = exp 2πim s l m l m=1

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 converges uniformly on compact subsets of D = s ∈ C : σ > 12 for almost all ω ∈ Ω with respect to the measure mH . Therefore, E(s; kl , α, ω) is an H(D)-valued random element defined on the probability space (Ω, B(Ω), mH ). Theorem 1.1. Suppose that Re α ≤ 0. Then the probability measure   k def τ PT (A) = νT E(s + iτ ; , α) ∈ A , A ∈ B(M(D), l converges weakly to the distribution of the random element E(s; kl , α, ω) as T → ∞. Note that the latter theorem implies the main result of [8]. Let s1 = 1, ( 1 + Reα if α 6= 0, s2 = 1 if α = 0, and

 2  Y 2s j f (s) = 1− s . 2 j=1

Then, clearly, f (s1 ) = 0 and f (s2 ) = 0, and s = 1 is a double zero of f (s) if α = 0. ˆ k , α) = f (s)E(s; k , α) is regular in D. Denote by |A| the Therefore, the function E(s; l l number of elements of a set A. Then, for σ > 1 (we consider the case Reα ≤ 0), ˆ k , α) = E(s; l =

 ∞   2  Y 2sj X σα (m) k 1− s exp 2πim = 2 ms l m=1 j=1

∞ X X

A⊆{1,2}

  P sj k σα (m) exp 2πim 2j∈A (−1)|A| 2−|A|s m−s = l m=1 =

2 X ∞ X

am,j

j=0 m=1



k ,α l



1 2js ms

,

where the coefficients am,j satisfy am,j ≪ σα (m), m ∈ N, j = 0, 1, 2.

2

A limit theorem for Dirichlet polynomials

Let n and N be positive integers, σ1 > 0 be fixed, and let
2 X N k n  m σ1 o X a k m,j l , α ˆN,n (s; , α) = E exp − , js ms l 2 n j=0 m=1 2

N

X X am,j ˆN,n (s; k , α, ω) = E l j=0 m=1

k , α ω j (2)ω(m) l 2js ms




n  m σ1 o exp − n

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583

ˆ k , α). Define two probability be two Dirichlet polynomials related to the function E(s; l measures   k PT,N,n (A) = νTτ EˆN,n (s + iτ ; , α) ∈ A , A ∈ B(H(D)), l and

PˆT,N,n (A) =

νTτ

  k EˆN,n (s + iτ ; , α, ω) ∈ A , l

A ∈ B(H(D)).

Lemma 2.1. The probability measures PT,N,n and PˆT,N,n both converge weakly to the same probability measure on (H(D), B(H(D))) as T → ∞. Proof of the lemma is similar to that of Lemmas 1 and 2 from [5].

3

A limit theorem for absolutely convergent series

Define, for positive integer n and σ1 > 12 ,
2 X ∞ k n  m σ1 o X a k m,j l , α Eˆn (s; , α) = exp − , l 2js ms n j=0 m=1 2

∞

X X am,j k Eˆn (s; , α, ω) = l j=0 m=1

k , α ω j (2)ω(m) l 2js ms




n  m σ1 o exp − . n

The latter two Dirichlet series both converge absolutely for σ > 12 . This can be obtained similarly to the case of the Riemann zeta-function, see [6], Chapter 5. In this section we will consider the weak convergence of probability measures   k τ PT,n (A) = νT Eˆn (s + iτ ; , α) ∈ A , A ∈ B(H(D)), l and PˆT,n (A) = νTτ

  k ˆ En (s + iτ ; , α, ω) ∈ A , l

A ∈ B(H(D)).

For the investigation of weak convergence of the above measures we need a metric on H(D) which induces its topology. It is well known, see, for example, [4], that there exists ∞ S a sequence {Kn } of compact subsets of D such that D = Kn , Kn ⊂ Kn+1 , and if K n=1

is a compact of D, then K ⊆ Kn for some n. Then ρ(f, g) =

∞ X n=1

2−n

ρn (f, g) , 1 + ρn (f, g)

f, g ∈ H(D),

where ρn (f, g) = sup |f (s) − g(s)|, is the desire metric. s∈Kn

Lemma 3.1. There exists a probability measure Pn on (H(D), B(H(D))) such that both the measures PT,n and PˆT,n converge weakly to Pn as T → ∞.

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Proof. By Lemma 2.1, the measures PT,N,n and PˆT,N,n both converge weakly to some measure PN,n as T → ∞. ˆ B(Ω), ˆ P) and Let θT be a random variable defined on a certain probability space (Ω, k ˆN,n (s + iθT ; , α). Then, by the uniformly distributed on [0, T ]. We put XT,N,n (s) = E l above remark, D XT,N,n −→ XN,n , (1) T →∞

where XN,n is an H(D)-valued random element with the distribution PN,n . By the Chebyshev inequality, for Mr > 0, P



sup |XT,N,n (s)| > Mr

s∈Kr



1 ≤ Mr T

ZT 0

k sup EˆN,n (s + iτ ; , α) dτ. l s∈Kr

The series for Eˆn (s; kl , α) converges absolutely for s ∈ D, therefore this shows that 1 1 lim sup P( sup |XT,N,n (s)| > Mr ) ≤ sup lim sup Mr N ≥1 T →∞ T T →∞ s∈Kr

ZT 0

≤ Ar < ∞.

ˆN,n (s + iτ ; k , α) dτ sup E l s∈Kr

(2)

r

Now let ε > 0 be an arbitrary number, and let Mr = Ar 2ε . Then from (2) we have lim sup P( sup |XT,N,n (s)| > Mr ) < T →∞

s∈Kr

ε 2r

(3)

for all r ∈ N = {1, 2, ...}. Since the function h : H(D) → R given by the formula h(f ) = sup |f (s)|, f ∈ H(D), is continuous, relation (1) yields s∈Kr

D

sup |XT,N,n (s)| −→ sup |XN,n (s)|. T →∞ s∈K r

s∈Kr

From this and (3) we find P( sup |XN,n (s)| > Mr ) ≤ s∈Kr

for all r ∈ N. Let us take Hε =



ε 2r

f ∈ H(D) : sup |f (s)| ≤ Mr , s∈Kr

(4)



r≥1 .

Then, clearly, Hε is a compact set of H(D), and, in view of (4), P(XN,n (s) ∈ Hε ) ≥ 1 − ε for all N ∈ N. Thus, we have shown that the family of probability measures {PN,n } is tight. Therefore, by Prokhorov’s theorem it is relatively compact. ˆN,n (s; k , α) and E ˆn (s; k , α) we have that lim EˆN,n (s; k , α) = By the definition of E l l l N →∞

Eˆn (s; kl , α), and since the series for EˆN,n (s; kl , α) absolutely converges for σ > convergence is uniform on compact subsets of D. Therefore, for every ε > 0,     k k τ ˆ ˆ lim lim sup νT ρ EN,n (s + iτ ; , α), En (s + iτ ; , α) ≥ ε ≤ N →∞ T →∞ l l

1 , 2

this

A. Laurinˇcikas / Central European Journal of Mathematics 3(4) 2005 580–590

1 ≤ lim lim sup N →∞ T →∞ εT

 ZT  k k ˆ ˆ ρ EN,n (s + iτ ; , α), En (s + iτ ; , α) dτ = 0. l l

585

(5)

0

Now let XT,n (s) = Eˆn (s + iθT ; kl , α). Then by (5) lim lim sup P (ρ (XT,N,n (s), XT,n (s)) ≥ ε) = 0.

N →∞

(6)

T →∞

Let {PN ′ ,n } ⊂ {PN,n } be a weakly convergent subsequence, say, to Pn , as N ′ → ∞. Then, D clearly, XN ′ ,n −→ Pn . The space H(D) is separable, therefore this (1), (6) and Theorem ′ N →∞ 4.2 from [1] show that D

XT,n −→ Pn , T →∞

(7)

that is, there exists a probability measure Pn such that PT,n converges weakly to Pn as T → ∞. Moreover, (7) shows that the measure Pn is independent of the choice of subsequence {PN ′ ,n }. The relative compactness of {PN,n } together with Theorem 2.3 of [1] gives the weak convergence of PN,n to Pn as N → ∞, and hence D

XN,n N−→ Pn . →∞

(8)

Now the above arguments applied to the random elements EˆN,n (s + iθT ; kl , α, ω) and Eˆn (s + iθT ; kl , α, ω), and (8) prove the weak convergence of PˆT,n to Pn as T → ∞. The lemma is proved. 

4

An approximation in the mean

ˆ k , α) we have to pass from Eˆn (s; k , α) to E(s; ˆ k , α). To prove a limit theorem for E(s; l l l For this an approximation of Eˆn (s; kl , α) in the mean is needed. Lemma 4.1. Let K be a compact subset of D. Then 1 lim lim sup N →∞ T →∞ T

ZT 0

ˆ + iτ ; k , α) − Eˆn (s + iτ ; k , α) dτ = 0. sup E(s l l s∈K

Proof. Let Γ(s) denote the Euler gamma-function, and let, for n ∈ N,   s s 1 ln (s) = Γ ns , σ1 > . σ1 σ1 2 Then, as in [8], we have, for σ > 12 , ˆn (s; k , α) = 1 E l 2πi

σZ 1 +i∞

ˆ + z; k , α)ln (z) dz . E(s l z

σ1 −i∞

(9)

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A. Laurinˇcikas / Central European Journal of Mathematics 3(4) 2005 580–590

  Now we will shift the line of integration in (9). Let σ belong to 12 + η, A , η > 0, when s ∈ K. We take σ2 = 21 + η2 . Since the integrand has a simple pole at the point z = 0, the residue theorem yields 1 k Eˆn (s; , α) = l 2πi

σ2 −σ+i∞ Z

ˆ + z; k , α)ln (z) dz + E(s; ˆ k , α). E(s l z l

(10)

σ2 −σ−i∞

Let L be a simple closed contour lying in D and enclosing K, and let δ denote the distance of L from the set K. Then by the Cauchy formula we have Z k k 1 ˆ ˆ E(z ˆ + iτ ; k , α) − E ˆn (z + iτ ; k , α) |dz|. sup |E(s + iτ ; , α) − En (s + iτ ; , α)| ≤ l l 2πδ l l s∈K L

Denoting by |L| the length of L, hence we find ZT

1 T

0

ˆn (s + iτ ; k , α) dτ ≪ ˆ + iτ ; k , α) − E sup E(s l l s∈K

|L| ≪ sup T δ σ+iu∈L

T Z+u u

E(σ ˆ + it; k , α) − Eˆn (σ + it; k , α) dt. l l

We can choose the contour L to satisfy the conditions inf {σ} ≥ s∈L

(10), we obtain

1 T

≪

−∞

≪

Z∞

−∞

+ 3η and δ ≥ η4 . Using 4

T Z+u u

Z∞

1 2

(11)

E(σ ˆ + it; k , α) − E ˆn (σ + it; k , α) dt ≪ l l

ln (σ2 − σ + iτ ) 1 T

TZ +u+τ u+τ

E(σ ˆ 2 + it; k , α) dtdτ ≪ l

 T +u+τ  12 Z dτ ˆ 2 + it; k , α) 2 dt . ln (σ2 − σ + iτ ) √  E(σ l T

(12)

u+τ

ˆ 2 + it; k , α) ≪ |E(σ2 + it; k , α)|, Lemma 3 of [8] yields the Since u is bounded and E(σ l l estimate O(T + |τ |) for the second integral in (12). Therefore, the left hand-side of (11) is ≪ sup

Z∞

σ+iu∈L −∞

ln (σ2 − σ + it) (1 + |t|)dt ≪

as n → ∞. The lemma is proved.

sup

Z∞

σ∈[−A,− η4 ] −∞

ln (σ + it) (1 + |t|)dt = o(1) 

A. Laurinˇcikas / Central European Journal of Mathematics 3(4) 2005 580–590

587

Let, for ω ∈ Ω and s ∈ D, ˆ k , α, ω) = E(s; l

2 X ∞ X

j=0 m=1

am,j



k ,α l



ω j (2)ω(m) . 2js ms

ˆ k , α, ω) is an H(D)-valued random element defined on the probability space Then E(s; l (Ω, B(Ω), mH ). Lemma 4.2. Let K be a compact subset of D. Then 1 lim lim sup n→∞ T →∞ T

ZT 0

for almost all ω ∈ Ω.

ˆ + iτ ; k , α, ω) − Eˆn (s + iτ ; k , α, ω) dτ = 0 sup E(s l l s∈K

ˆ k , α, ω) it follows that, for σ > 1 , Proof. From Lemma 5 of [8] and definition of E(s; l 2 ZT

ˆ + it; k , α, ω)|2dt ≪ T |E(σ l

0

for almost all ω ∈ Ω. Using this, to prove the lemma it suffices to repeat the arguments of the proof of Lemma 4.1. 

5

ˆ k , α) A limit theorem for the function E(s; l

In this section we will consider the probability measures   k τ ˆ QT (A) = νT E(s + iτ ; , α) ∈ A , A ∈ B(H(D)), l and ˆ T (A) = ν τ Q T

  k ˆ E(s + iτ ; , α, ω) ∈ A , l

A ∈ B(H(D)).

Lemma 5.1. There exists a probability measure P on (H(D), B(H(D))) such that both ˆ T converge weakly to P as T → ∞. the measures QT and Q ˆ k , α) and E(s; ˆ k , α, ω) Proof. We use the same method as in the proof of Lemma 2 with E(s; l l ˆn (s; k , α) and Eˆn (s; k , α, ω), respectively, and with Eˆn (s; k , α) and E ˆn (s; k , α, ω) instead of E l l l l ˆN,n (s; k , α, ω), respectively. For this we apply Lemmas 3.1, instead of EˆN,n (s; k , α) and E l

4.1 and 4.2.

l



Lemma 5.2. The probability measure QT converges weakly to the distribution of the ˆ k , α, ω) as T → ∞. random element E(s; l

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A. Laurinˇcikas / Central European Journal of Mathematics 3(4) 2005 580–590

Proof. It remains to prove that the limit measure P in Lemma 5.1 coincides with the ˆ k , α, ω). distribution of the random element E(s; l Let A ∈ B(H(D)) be a continuity set of P . Then by Lemma 5.1   k τ ˆ + iτ ; , α, ω) ∈ A = P (A) lim νT E(s (13) T →∞ l for almost all ω ∈ Ω. We fix the set A and define a random variable θ on (Ω, B(Ω), mH ) by ( ˆ k , α, ω) ∈ A, 1 if E(s; l θ(ω) = k ˆ 0 if E(s; l , α, ω) ∈ / A. Denote by Eξ the expectation of ξ. Then we have Z ˆ k , α, ω) ∈ A) = P ˆ (A) < ∞, Eθ = θdmH = mH (ω ∈ Ω : E(s; E l

(14)

Ω

ˆ k , α, ω). where PEˆ is the distribution of E(s; l Let, for τ ∈ R, aτ = {p−iτ : p is prime }. Then {aτ : t ∈ R} is a one-parameter group. Define ϕτ : Ω → Ω by ϕτ (ω) = aτ ω, ω ∈ Ω. This gives a one-parameter group {ϕτ : τ ∈ R} of measurable transformations on Ω. In [6] it was obtained that the latter group is ergodic. Hence we have that the process θ(ϕτ (ω)) is also ergodic. Therefore, by the classical Birkhoff-Khintchine theorem 1 lim T →∞ T

ZT

θ(ϕτ (ω))dτ = Eθ

(15)

0

for almost all ω ∈ Ω. However, on the other hand, 1 T

ZT 0

θ(ϕτ (ω))dτ =

νTτ

  k ˆ E(s + iτ ; , α, ω) ∈ A . l

Therefore, in view of (14) and (15)   k τ ˆ + iτ ; , α, ω) ∈ A = P ˆ (A) lim νT E(s E T →∞ l for almost all ω ∈ Ω. Thus, P (A) = PEˆ (A) by (13) for every continuity set A of P . Hence P (A) = PEˆ (A) for all A ∈ B(H(D), and the lemma is proved. 

6

A two-dimensional limit theorem

Let H 2 (D) = H(D) × H(D). Define on (Ω, B(Ω), mH ) an H 2 (D)-valued random element F (s, ω) by the formula   k ˆ F (s, ω) = f (s, ω), E(s; , α, ω) , l

A. Laurinˇcikas / Central European Journal of Mathematics 3(4) 2005 580–590

where f (s, ω) =

2  Y j=1

589

 2sj ω(2) 1− . 2s

Lemma 6.1. The probability measure   k def τ ˆ PT,f,Eˆ = νT (f (s + iτ ), E(s + iτ ; , α)) ∈ A , l

A ∈ B(H 2 (D)),

converges weakly to the distribution of the random element F (s, ω) as T → ∞. Proof. The function f (s) is a Dirichlet polynomial. Therefore, the probability measure νTτ (f (s + iτ ) ∈ A), A ∈ B(H(D)), converges weakly to the distribution of the random element f (s, ω) as T → ∞. This and Lemma 5.2 by a standard method (a modification of the Cram´er-Wald criterion), see, for example, [7], yield the assertion of the lemma. 

7

Proof of the Theorem 1.1

The main result of this paper is a simple consequence of Lemma 6.1. Proof (of Theorem 1.1). Define a function h : H 2 (D) → M(D) by theformula  g2 1 1 h(g1 , g2 ) = g1 , g1 , g2 ∈ H(D). The metric d has the property d(g1 , g2) = d g1 , g2 , therefore, the function h is continuous. Consequently, the measure PT = PT,f,Eˆ h−1 converges weakly to the measure ! ˆ k , α, ω) E(s; l ∈ A , A ∈ B(M(D)), (16) mH ω ∈ Ω : f (s, ω) as T → ∞. However, ˆ k , α, ω) = E(s; l

2 X ∞ X

j=0 m=1

am,j



k ,α l



ω j (2)ω(m) = 2js ms

 ∞   2  Y 2sj ω(2) X σα (m)ω(m) k exp 2πim = = 1− s s 2 m l m=1 j=1

k = f (s, ω)E(s; , α, ω). l This and (16) prove the theorem.



References [1] P. Billingsley: Convergence of Probability Measures, Wiley, New York, 1968. ¨ [2] H. Bohr and B. Jessen: “Uber die Wertverteilung der Riemannschen Zeta funktion”, Erste Mitteilung, Acta Math., Vol. 54, (1930), pp. 1–35.

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¨ [3] H. Bohr and B. Jessen: “Uber die Wertverteilung der Riemannschen Zeta funktion”, Zweite Mitteilung, Acta Math., Vol. 58, (1932), pp. 1–55. [4] J.B. Conway: Functions of One Complex Variable, Springer-Verlag, New York, 1973. [5] J. Genys and A. Laurinˇcikas: “Value distribution of general Dirichlet series. IV”, Liet. Matem. Rink., Vol. 43, No 3, (2003), pp. 342–358; Lith. Math. J., Vol. 43, No 3, (2003), pp. 281–294 (in Russian). [6] A. Laurinˇcikas: Limit Theorems for the Riemann Zeta-function, Kluwer, Dordrecht, Boston, London, 1996. [7] A. Laurinˇcikas and R. Garunkˇstis: The Lerch Zeta-Function, Kluwer, Dordrecht, Boston, London, 2002. [8] A. Laurinˇcikas: “Limit theorems for the Estermann zeta-function. I”, Statist. Probab. Letters, Vol. 72(3), (2005), pp. 227–235. [9] M. Lo`eve: Probability Theory, Van Nostrand, Toronto, 1955.

CEJM 3(4) 2005 591–605

Tchebotar¨ ov’s extremal problem Promarz M. Tamrazov∗ Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine

Received 4 April 2005; accepted 28 June 2005 Abstract: We give the complete solution of the extremal problem posed by N.G. Tchebotar¨ ov in 20th of the last century, and we establish explicit parametric formulae for the extremals. c Central European Science Journals. All rights reserved.

Keywords: Tchebotar¨ ov’s problem, conformal mappings, logarithmic capacity, quadratic differentials, Riemann surfaces, graphs MSC (2000): 30(C,D,E), 31(A,B,C), 33E, 39B

1

Introduction

In 1929 G. Polya [10] discussed the extremal problem earlier posed by N. G. Tchebotar¨ov. We formulate it in the well-known equivalent form: among all univalent conformal mappings f of the unit disk K := {z ∈ C : |z| < 1} of the complex plane C into the plane C punctured at a finite number of fixed points a1 , . . . , am ∈ C \ {0}, with f (0) = 0, to find such an f for which the functional |f ′ (0)| achieves its maximal value. In such a form Tchebotar¨ov’s problem will be called inner (unlike another completely equivalent form called outer, in which the minimization of logarithmic capacity of continua in C containing a given finite collection of fixed finite points is discussed). First essential results in the Tchebotar¨ov’s problem were obtained by M. A. Lavrentiev [7; 8] and H. Gr¨otzsch [3] in 1930. Later on G. M. Goluzin [1; 2, p. 152-157] further developed these investigations. Besides existence and uniqueness of the extremal function f, these authors also established some qualitative and structural properties of the extremal ∗

E-mail: [email protected]

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P.M. Tamrazov / Central European Journal of Mathematics 3(4) 2005 591–605

and the following functional-differential equation for it (see [7; 8; 3; 1; 2, p. 152-157]): (

zf ′ (z) 2 p(f (z)) ) = , f (z) q(f (z))

where p(w) :=

Ym

j=1

(1)

(aj − w),

and q is a polynomial on w ∈ C of the degree m − 1 with Ym q(0) = aj . j=1

¯ (K) Moreover, f is regular on ∂K except at a finite number of points, and the set B := C\f is connected and is a union of a finite number of (open) analytic arcs and their endpoints. It follows that the domain f (K) is admissible with respect to the quadratic differential Q(w)dw 2 := −

q(w) dw 2 2 w p(w)

(2)

(for terminology and main facts concerning theory of quadratic differentials, see [4]). But the equation (1) contains m − 1 complex-valued parameters (coefficients of q) whose values were unknown, and the problem of finding explicit formulae for the extremals of the Tchebotar¨ov‘s problem remained unsolved (see [2, p. 156; 11, p. 202]). In 1965 G.V. Kuz’mina [6], in the particular case of this problem with m = 2 (which corresponds to the case with a single unknown parameter), obtained some implicit expression for the extremal. (By the way, the functional-differential equation (1) is the source of the great idea – the important role of quadratic differentials in complex analysis and its applications.) Note that because of unknown parameters in (1), the integration of (1) gives no benefit for finding formulae for extremals of the Tchebotar¨ov’s problem. Moreover such a situation is common for extremal problems of complex analysis. Up to now formulae for extremals were known only for some problems under special conditions: (1) When a quadratic differential has a single in C pole. (Here we mention the wellknown Bieberbach problem on coefficients of univalent functions, in which the number of unknown parameters equals the order of the mentioned pole minus 2, and this is why the problem required many decades before has been solved by de Branges). (2) If the number of poles is greater than 4 and their location is free of any additional restrictions, then formulae for extremals were known only in some cases with one or at most two unknown parameters in the corresponding quadratic differentials. In the present work the author gives the general and complete solution of the above mentioned open problem – we establish formulae for extremals of the Tchebotar¨ov’s problem in general case – for any integer m ≥ 2 (and hence with any number of arbitrarily distributed poles and any number of unknown parameters in the corresponding quadratic differential). These formulae are explicit (see formulae (3) and Theorem 4.3 below), and it is a consequence of an extremely surprising fact that the derivative of the extremal

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(which is equal to the integrand in (3)) is a product of simple factors representing separate singularities and zeros of the derivative. Moreover geometric properties of functions given by (3) are of great importance irrespective of the Tchebotar¨ov’s problem, see Theorem 4.1. In 2004 this article was published in the preprint form [13]. I am thankful to Professor P. Lelong for the interest to results of this work and their proof, and also for presenting for publication in Comptes Rendus Mathematique the note [14] announcing results of [13]. On 14 December 2004 the results of [13] were reported in Kiev at the Scientific Council of the Institute of Mathematics of the National Academy of Sciences of Ukraine. We emphasize that among numerous extremal problems generating quadratic differentials with a number of arbitrarily distributed fixed poles, it is the first case when the extremals are found in explicit form. And our methods enable also to solve a series of other extremal problems of the analogous nature. Part of this work was carried out during the author’s visit to the Gebze Institute of Technology, and the author is greatly appreciated to Professor Alinur B¨ uy¨ ukaksoy for the kind invitation and hospitality.

2

Normalized formulation of the problem

¯ the quadratic differential Q(w)dw 2 has one double pole at the origin and at least On C ¯ \ {0}. Therefore (see [4, p. 62]) it has a simple pole at least in two simple poles on C one of the points aj ∈ C \ {0}. Since the formulation of the inner Tchebotar¨ov‘s problem waj is invariant with respect to the transformation w 7→ aj −w with aj 6= ∞, it follows that without any loss of generality, we may assume that Q(w)dw 2 has simple poles at the point am+1 = ∞ and at least at some finite point aj ∈ B. If some of (finite) the points aj are regular points or zeroes of Q(w)dw 2, then without loss of generality we may omit them from the consideration (see [8]). So we assume that all points a1 , . . . , am+1 are simple poles of Q(w)dw 2, where am+1 (= ∞) is distinguished, while the collection a1 , . . . , am is unordered. Such a point collection {aj } := {aj }m+1 j=1 will be called normalized. All points of a normalized collection are endpoints of the set B (see [7; 8]). Every other point of B is either a zero of Q(w)dw 2 (and then it is a branch point of B), or this point is a regular, non-critical point of Q(w)dw 2 (and then it belongs to a trajectory of this quadratic differential). Thus we ¯ consisting may consider B as an undirected, connected, simple, acyclic, plane graph on C of nodes of order one at all points aj and only in them, nodes of orders νj + 2 at all zeroes bs of degrees νs ≥ 1, and only in them, and of all analytic trajectories of Q(w)dw 2 (contained in B and ending at zeros or simple poles of Q(w)dw 2) as edges of the graph. This curvilinear geometric graph is connected and has no cycles. Consequently it is a tree, and we shall denote it by L({aj }). The total multiplicity of all zeroes of Q(w)dw 2 equals m − 1. Let now k be the number of different zeroes of Q(w)dw 2. Then the number of edges of the graph under consideration is m + k (on the basis of the Euler’s theorem).

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The particular case m = 1 of the Tchebotar¨ov‘s problem is a well-known K¨obe‘s covering theorem on 1/4. Therefore for simplicity of formulations in the sequel we shall exclude this case and assume that m ≥ 2.

3

Construction of extremals

Now we shall construct a class of explicitly defined functions containing all extremals for the inner Tchebotar¨ov‘s problem and only them. This class will be parametrized by means of special geometric rectilinear graphs defined in the complex plane. Let G be the class of all finite, undirected, connected, simple plane graphs Γ each of which satisfies the following conditions: (1) each edge γ of Γ is a rectilinear open interval in C of the length |γ| > 0, and these intervals mutually do not intersect each other, while nodes of the graph coincide with the endpoints of these intervals; (2) Γ does not contain nodes of order 2 and cycles; (3) the sum of lengths |γ| of all intervals γ of the graph Γ equals π; (4) the point ζ = 0 is a node of Γ of order 1, and the edge of Γ incident to this point is contained in the real half-axis Re ζ > 0. Let Supp Γ denote the closure in C of the geometric union of all edges of the graph Γ ∈ G. Starting at the node 0, let us run along Γ in the direction in which the complementary to Γ domain C \ (Supp Γ) remains on the left. Such a pass of Γ will be called natural. For every point ζ on an edge γ ∈ Γ, let r1 (Γ, ζ) and r2 (Γ, ζ) denote the length of the pass respectively to the first and to the second reaching the point ζ, while r1 (Γ, 0) = 0, r2 (Γ, 0) = 2π. Under a single such pass along an edge γ the growth of each of functions r1 and r2 equals |γ|. For every node v of the order τ (v), let r1 (Γ, v), . . . , rτ (v) (Γ, v) denote the length from the start till 1st, . . . , τ (v)th pass of v. For every ζ ∈ Supp Γ and all j = 1, . . . , τ (ζ) let us denote εΓ,j (ζ) := exp (irj (Γ, ζ)). Let Γ′ be one more graph from G, and for every ζ ′ ∈ Supp Γ′ the objects τ ′ (ζ ′ ) and εΓ′ ,j (ζ ′ ) are defined exactly as analogous objects were defined for Γ and ζ ∈ Supp Γ. Then the graphs Γ and Γ′ will be called equivalent, if there exists the isomorphism η : Γ → Γ′ such that η(0) = 0 and for every node v of Γ we have rj (Γ′ , η(v)) = rj (Γ, v) ∀j = 1, . . . , τ (v). If graphs Γ, Γ′ ∈ G are equivalent, then for every ζ ∈ Supp Γ there corresponds a uniquely defined ζ ′ ∈ Supp Γ′ for which εΓ,j (ζ) = εΓ′ ,j (ζ ′ ) ∀j = 1, . . . , τ (ζ). For a graph Γ, let V (Γ) be the set of all its nodes of order 1, and W (Γ) be the set of all other its nodes (of orders ≥ 3 ). Let V be the set of all points εΓ,1 (p) (∈ T ), when p runs

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through the set V (Γ). Denote by Wv the set of all points εΓ,j (v) (∈ T ), when v ∈ W (Γ) is fixed and j runs through the set of values 1, . . . , τ (v). Denote also W := ∪v∈W (Γ) Wv . Clearly the point z = 1 is contained in V . Under assumptions of Section 2, the cardinal numbers of the sets V (Γ)\{1} and W (Γ) are m and k, respectively, and Γ contains exactly m + k edges. ¯ \ (W ∪ {1}), With any fixed branch of the below integrand continuous at the set K ¯ let us consider the function for z ∈ K f (z) :=

Zz

(ζ − 1)−3 (

0

Y

α∈V \{1}

(ζ − α))

Y

v∈W (Γ)

(

Y

β∈Wv

(ζ − β)2−τ (v) )1/τ (v) dζ.

(3)

We have |f ′(0)| = 1. Let fK denote the restriction of f to K.

4

Main results

For a fixed graph Γ ∈ G under the above notations and assumptions of Section 2, we get the following result. Theorem 4.1. The function f given by (3) is holomorphic and univalent in K, contin¯ ¯ \ {1}, continuous in the generalized sense (with respect to the topology of C uous in K ¯ For every point ζ0 ∈ Γ the function f glues rational-analytically in the image) on K. all points εΓ,j (ζ0 ) (j = 1, . . . , τ (ζ0 )) into one point denoted by y(ζ0), and f is contin¯ \W uously and meromorphically extendable into a neighbourhood of every point z ∈ K ¯ = C, ¯ f (0) = 0, f (1) = ∞, and (holomorphically for every z 6= 1). Moreover f (K) the function fK is extremal in the inner Tchebotar¨ ov’s problem for the collection of all points a(p) := y(p) where p runs over the whole set V (Γ). The extremal function in this problem for the mentioned collection of points a(p) is unique up to rotation of the disc K in the z-plane around the origin. The set of all simple poles of the quadratic differential Q(w)dw 2 given by (2) is normalized and hence coincides with the set of all m + 1 points a(p), including y(0) = ∞, while the set of all zeroes of Q(w)dw 2 coincides with the set of all points b(v) := y(v), where v runs over all k points of the set W (Γ). Each point a(p) (including a(0) = ∞) is an endpoint of some single trajectory of Q(w)dw 2. The boundary ¯ is the union of m + k trajectories of Q(w)dw 2, their of the domain f (K) with respect to C m + 1 endpoints a(p) (∀p ∈ V (Γ)) and k points b(v) (∀v ∈ W (Γ)). Let Γ ∈ G be the fixed graph from Theorem 4.1 with all related objects and notations (in particular, a(p) for all V (Γ)). Using notations of Section 2, denote L({a(p)}) =: Γ∗ . Then we get the following result.

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Theorem 4.2. The graph Γ∗ is isomorphic to Γ, with the correspondence of the node ζ = 0 of Γ to the node w = ∞ of Γ∗ , and the pass of Γ∗ in the direction in which the ¯ \ Γ∗ remains on the left, corresponds to the pass of Γ in the natural direction domain C (see Section 3). Then the length of every pass along Γ∗ in the metric |Q1/2 dw| equals the length of its pre-image on Γ with respect to the natural length measuring on Γ (see Section 3). Thus the graphs Γ and Γ∗ are isomorphic, equally oriented relative to their comple¯ domains and isometric in the sense of Theorem 4.2 (this mentary (with respect to C) isometry being consistent with the isomorphism and the direction of pass). From the definitions we see that for any equivalent graphs Γ′ , Γ′′ ∈ G and related to them objects corresponding to each other in this equivalence (including objects of the form p, v, εΓ,1 (p), εΓ,j (v), V (Γ), W (Γ) for these graphs), the objects V, W, τ (v), Wv , a(p), b(v), f of similar form coincide. ˜ denote the factor-set of G with respect to the equivalence. Let G For a graph Γ ∈ G, let {Γ} denote the class of all graphs from G equivalent to Γ. Let N denote the set of all normalized point collections. If the function z 7→ h(z) is extremal in the inner Tchebotar¨ov’s problem for the collection {aj } ∈ N, then under every c ∈ C \ {0} and t ∈ T the function z 7→ ch(tz) is extremal in the inner Tchebotar¨ov’s problem for the collection {caj } and there exists (the unique) t0 ∈ T such that h(t0 ) = ∞. Therefore for every {aj } ∈ N there exists (the unique) extremal h0 in the inner Tchebotar¨ov’s problem for the collection {aj } such that h0 (1) = ∞. Without loss of generality we may consider only the case when h0 (1) = ∞. ˜ → N be the mapping defined for each Γ ˜ as the collection {f (p)}p∈V (Γ) , Let H : G ˜ and the corresponding V (Γ). where f is the function (3) defined for arbitrary Γ ∈ Γ Theorem 4.3. The class of all extremals of the Tchebotar¨ ov‘s problem is parametrized by ˜ elements of the set G and a positive number r, and this parametrization is a one-to-one correspondence: ˜∈G ˜ there corresponds one (and only one) normalized collection of (1) to every element Γ points for which the function fK with f given by (3) and corresponding to each graph ˜ is extremal in the inner Tchebotar¨ Γ∈Γ ov‘s problem; and here we have |f ′(0)| = 1; (2) and conversely, for every point collection {aj } ∈ N there exists one and only one ˜ ∈G ˜ and the unique positive constant r such that H(Γ) ˜ = {aj /r} and the class Γ ˜ is extremal in the inner function rfK with f defined by (3) for arbitrary Γ ∈ Γ Tchebotar¨ov‘s problem for {aj }. In particular, for m ≤ 2 from our results we get that the restriction to K of the function Zz (ζ + eiδ2 )(ζ + e−iδ3 )dζ f (z) := (ζ − 1)3 [(ζ 2 − 2ζ cos δ1 + 1)(ζ + ei(δ2 −δ3 ) )]1/3 0

with any constants δ1 > 0, δ2 > 0, δ3 ≥ 0, under δ1 + δ2 + δ3 = π, is extremal

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in the inner Tchebotar¨ov‘s problem for the collection of points a1 = f (ei(δ1 +δ2 ) ), a2 = f (e−i(δ1 +δ3 ) ), a3 = f (1) = ∞; and |f ′ (0)| = 1. Under the additional requirement that δ3 6= 0, the point collection {a1 , a2 , a3 } is normalized (corresponding to m = 2). To compare this particular case of our result, we mention that for the case m = 2 G. V. Kuz’mina [6] expressed the extremal as an implicit solution of a system of equations containing elliptic Jacobi functions. The further sections are devoted to the proof of the above results.

5

Couples of arcs and stars

Let ∆ denote the class of all (unordered) couples {δ + , δ − } consisting of non-intersecting open arcs δ + , δ − on the unit circle T ⊂ C. For a couple {δ + , δ − } =: δ ∈ ∆ let us denote < δ >:= δ + ∪ δ − . This < δ > will be called the support of δ. Let also |δ + | and |δ − | be lengths of δ + and δ − , respectively. Let ∆0 be the set of all δ := {δ + , δ − } for which the closure Clos < δ > of the set < δ > is connected and does not coincide with T, and in this case let P (δ) denote the common endpoint of δ + and δ − . For any n ∈ N, an unordered collection {δ1 , . . . , δn } =: A will be called one-sheeted, if < δ1 >, . . . , < δn > are mutually non-intersecting. We say that δ1 , . . . , δn are members of A. The set ∪nk=1 < δk >=: supp A will be called the support of A. A one-sheeted unordered collection A = {δ1 , . . . , δn } will be called a star, if every member δj of A has a connected component S(δj , A) of the set T \ < δj > which contains supports of all other members of A. In such a case the other connected component of T \ < δj > will be denoted by S0 (δj , A) and called the shadow of δj with respect to A. The set ∪nk=1 S0 (δj , A) =: S0 (A) will be called the shadow of A. A star A will be called connectable, if the set T \ (S0 (A) ∪ supp A) consists of a finite number of points.

6

Constellations

A finite, non-empty collection of connectable stars will be called a constellation. A constellation C will be called acyclic, if each δ ∈ ∆ is a member of at most two stars from C and < δ > does not intersect supports of other stars from C. A constellation C will be called irreducible, if every star in C contains at least three members. Let A1 and A2 be different stars from C, and δ be a member of both of these stars. Then A1 and A2 will be called neighbours, and δ will be called a link in C (between these neighbours).

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Let C be an acyclic, irreducible constellation. Every two neighbours from C have only one link between them in C. Let k ≥ 1 be an integer. A sequence δ1 , . . . , δk of links in C is called a chain (of the length k) in C, if there exists a sequence A1 , . . . , Ak+1 of stars in C with the following properties: (1) Aj 6= Aj+1 and δj is the link between Aj and Aj+1 for each j = 1, . . . , k; (2) if k ≥ 2, then also δj 6= δj+1 for all j = 1, . . . , k − 1. The stars A1 , . . . , Ak+1 are called vertices of the chain δ1 , . . . , δk . From our definitions and assumptions there follows that for every chain of the length k ≥ 2 all links of the chain are mutually different, and all vertices A1 , . . . , Ak+1 of the chain are mutually different as well. A chain δ1 , . . . , δk in C will be called maximal, if it is not contained in a longer chain in C (this means that there is no link δ in C such that some of the sequences δ, δ1 , . . . , δk or δ1 , . . . , δk , δ is a chain in C ). Every chain in C is contained in a maximal chain in C. If a sequence δ1 , . . . , δk in C is a maximal chain in C and A1 , . . . , Ak+1 are the vertices of this chain, then all members δ0 of A1 different from δ1 , and all members δk+1 of Ak+1 different from δk satisfy the following condition: their shadows S0 (δ0 , A1 ) and S0 (δk+1 , Ak+1 ) do not intersect supports of other stars from C. Every A ∈ C containing at most one link will be called a margin star of C. It is easily verified that C contains at least one margin star. We see also that a margin star has at most one neighbour star. A constellation C will be called connected if for each two different (if any) stars A′ , A′′ from C there exists a chain δ1 , . . . , δk in C such that δ1 is a member of A′ and δk is a member of A′′ .

7

Germs of stars and flatting sequences of constellations

Let us fix an acyclic, irreducible, connected constellation C. Let A1 be a margin star from C. If C contains µ ≥ 2 stars, then A1 contains a link δ1 := {δ1+ , δ1− } ∈ ∆ to its neighbour star A2 (which is uniquely defined). We shall modify C and construct some new constellation C ′ with the following properties: (i) We exclude the star A1 from C; (ii) We replace A2 := {δ1 , δ2 , . . . , δn } (n ≥ 3) by another connectable star {δ, δ2 , . . . , δn } =: A2∗ where δ = {δ + , δ − } ∈ ∆ and δ + , δ − are contained in δ1+ , δ1− , respectively, with |δ + | = 21 |δ1+ |, |δ − | = 12 |δ1− |. (iii) We keep in C ′ all other stars contained in C (except A1 , A2 ). We say that C ′ is cut from the margin star A1 ∈ C across its member δ1 . One can verify that the new constellation C ′ is also acyclic, irreducible, connected and it contains µ − 1 stars (less than C ). Denote by (A1 )∗ =: A1∗ the connectable star obtained from A1 by means of replacing δ1 = (δ1+ , δ1− ) by the member δ∗ = (δ∗+ , δ∗− ) ∈ ∆ with δ∗+ , δ∗− contained in δ1+ , δ1− , respec-

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tively, with |δ∗+ | = 12 |δ1+ |, |δ − | = 21 |δ1− |. The star (A1 )∗ =: A1∗ will be called the germ of A1 . If C ′ contains more than one star, we may proceed in the same style and construct a sequence of acyclic, irreducible, connected constellations C1 := C, C2 := C ′ , . . . , Cµ := ′ Cµ−1 such that Cµ consists of a single star A(µ) which contains at least three members (µ) δj , and for every ν = 1, . . . , µ − 1 the constellation Cν+1 is cut from some margin star A(ν) ∈ Cν across some its member δ (ν) . Such a sequence C1 := C, C2 := C ′ , . . . , Cµ := ′ Cµ−1 of constellations will be called flatting. If C contains only one star A1 , then we denote A1∗ := A1 and then the flatting sequence consists of the single constellation C.

8

Sewing and construction of an orientable compact Riemann surface

Let ∆1 be the class of all δ = (δ + , δ − ) ∈ ∆ for which |δ + | = |δ − |. Denote also ∆10 := ∆0 ∩ ∆1 . For δ := (δ + , δ − ) ∈ ∆1 let hδ : δ + → δ − be the isometric homeomorphism between the arcs δ + and δ − which satisfies the condition Re zh′δ (z)/hδ (z) = −1. This homeomorphism will be called δ-admissible. Every two points of < δ > related to each other in the homeomorphism hδ will be called δ-corresponding. Let δ := {δ + , δ − } ∈ ∆1 and the set < δ > be symmetric with respect to the ray arg z = const = θ. In the case when δ + and δ − have a common endpoint P (δ), we set θ = arg P (δ). Let us consider the map η : z 7→ (

ze−iθ − 1 2 ) . zeiθ + 1

It maps K onto the plane C cut along the interval (−∞, 0] ⊂ R. This map glues every couple of points z + and z − which are δ-corresponding. With respect to every couple (z + , z − ) =: z˘ of δ-corresponding points of < δ >, let us introduce one point zˆ and denote by Kδ the set-theoretic union of K and the set of all mentioned points zˆ. Let us add the boundary set η(δ + ) = η(δ − ) to η(K) and get the domain denoted by ηδ (K). Then we extend the Riemann surface K equipping the set Kδ at every its point zˆ with the local topological and conformal structure inherited from ηδ (K). As a result, we get a Riemann surface denoted also by Kδ and being an extension of K. Clearly Kδ is orientable and schlichtartig. If moreover δ ∈ ∆10 , then we add the point w = 0 to ηδ (K) and get a domain denoted by η δ (K). Now with respect to P (δ) we introduce the point Pˆδ , add it to Kδ and equip the resulting set union K δ with the local topological and conformal structure inherited

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from η δ (K). The resulting Riemann surface which is an extension of Kδ will be denoted by K δ as well. It is orientable, schlichtartig and simply connected. Now let us define the point-theoretic union of such Riemann surfaces corresponding to all δ ∈ A and all A ∈ C : K(A) := (∪δ∈A∩∆1 Kδ ) ∪ (∪δ∈A∩∆10 K δ ), K(C) := ∪A∈C K(A). Let us fix a star A = {δ1 , . . . , δn } (n ≥ 3) from C and enumerate shadows Sj of δj (j = 1, . . . , n) with respect to A in their natural counterclockwise order on T . For all integers j and k we assume δj+kn := δj , Sj+kn := Sj . Denote Sj ∪ < δj >=: Hj , and let αj be the common endpoint of Hj−1 and Hj . For every j we assume that δj = (δj+ , δj− ), + while αj is an endpoint both of δj− and δj−1 . Also let P − (δj ) be the end of δj− other than αj , and P + (δj ) be the end of δj+ other than αj+1 . For every j consider the sets + Yj := δj− ∪ δj−1 ∪ {αj },

Dj := {z ∈ C \ {0} :

z ∈ Yj } |z|

and any continuous in Dj branch of the function ψj : Dj → C defined by the formula ψj (z) := (log

z 2/n ) αj

and normalized by the condition arg ψj (z) = −

2πj n

∀z ∈ δj− .

Then 2π(j − 1) + ∀z ∈ δj−1 , n 2πj arg ψj (z) = arg ψj+1 (z) = − ∀z ∈< δj >, n ψj (αj eiτ ) = ψj+1 (αj+1 e−iτ ) ∀τ ∈ (0, |δj+ |). arg ψj (z) = −

(4)

The sets Dj are mutually disjoint. Now let us consider the set D := ∪j Dj and the function ψ : D → C given by the formula ψ(z) := ψj (z) ∀z ∈ Dj (j = 1, . . . , n). (5) ˆ D.

It is univalent, holomorphic and has continuous extension ψˆ onto the set D ∪(∪j Yj ) =:

Moreover because of (4), for every δj the function ψˆ glues the arcs δj+ and δj− into one inner line δˆj point-wise and isometrically (each couple of δj -corresponding points of < δj > is mapped into one inner point which is the image of the appropriate point

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of K(A) \ K). By (5), ψˆ glues all points α1 , . . . , αn into the point 0. So with respect to the star A, we may extend the Riemann surface K(A) adding one point denoted by α(A) to K(A) and equipping the resulting set R(A) at the point α(A) with the local ˆ ˆ This R(A) with topological and conformal structure inherited from the ψ-image of D. such a structure is an orientable and schlichtartig Riemann surface, and we denote it by R(A). Now let us define the Riemann surface which is the extension of all such R(A) : R(C) := ∪A∈C R(A). Obviously it is orientable and compact. ¯ there corresponds the unique point of R(C) (obtained by So for every point z ∈ K means of the above procedures of sewing) which will be denoted by ηC (z). The function ¯ onto R(C). ηC maps K

9

Exhaustion by simply connected Riemann surfaces and schlichtartigkeit

Let C be an acyclic, irreducible, connected constellation, and C1 := C, C2 := C ′ , . . . , ′ Cµ := Cµ−1 be the corresponding flatting sequence of constellations. Then there exists some sequence of margin stars A(1) ∈ C1 , . . . , A(µ) ∈ Cµ with links δ (1) ∈ A(1) , . . . , δ (µ−1) ∈ A(µ−1) such that Cν+1 is cut from A(ν) across δ (ν) for all ν = 1, . . . , µ − 1. We start with K and consider a margin star A(1) = {δ1 , . . . , δn } from C for which δ1 is a link. Then all other members of A(1) belong to ∆10 and K δ2 is simply connected. Denote by K δ2 ,δ3 the Riemann surface obtained by adding to K δ2 the set < δ3 > and the point P (δ3 ) and equipping this union with the local topological and conformal structure (at the added set) inherited from R(A(1) ). This K δ2 ,δ3 is also simply connected. We continue the extension of the resulting Riemann surfaces with respect to δj for j = 3, . . . , n and consequently construct simply connected Riemann surfaces K δ2 , ..., δj , . . . , K δ2 , ..., δn . Then let us construct the Riemann surface F (K, A(1) , C) obtained by adding to K δ2 , ..., δn the set < δ1 > and the point α(A(1) ) and equipping the resulting set with the local topological and conformal structure (at the added set) inherited from R(A(1) ). Of course (1) (1) we have F (K, A(1) , C) = R(A(1) ). Now let A∗ be the germ of A(1) . Let F (K, A∗ , C) be (1) the Riemann surface obtained in analogy to F (K, A(1) , C) with the star A∗ instead of (1) A(1) . It also is simply connected and F (K, A∗ , C) ⊂ F (K, A(1) , C). So if C contains only one star, then R(C) is simply connected. (2) Now let µ > 1, A(2) be a margin star from C2 , and A∗ be the germ of A(2) (defined (1) like A∗ above). Starting from the simply connected Riemann surface F1 := K (1)

and the margin star A∗ ∈ C1 , we get the simply connected Riemann surface (1) F2 := F (F1 , A(1) ∗ , C1 ) = R(A∗ )

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P.M. Tamrazov / Central European Journal of Mathematics 3(4) 2005 591–605 (1)

which is an extension of F1 ∪ R(A∗ ). If µ > 2, then we deal with the simply connected Riemann surface F2 , consider the (2) margin star A∗ ∈ C2 and construct the simply connected Riemann surface F3 := F (F2 , A(2) ∗ , C2 ) (2)

which is an extension of F2 ∪ R(A∗ ). This is shown in the same way as the preceeding statements. Continuing analogous arguments, we construct a sequence of Riemann surfaces F1 , . . . , Fµ with the following properties: for all ν = 1, . . . , µ−1 they are simply connected and Fν ∪ R(A(ν) ∗ ) ⊂ Fν+1 , (µ)

and since Cµ contains only one star A∗ , as above we get that Fµ is simply connected as well. Obviously Fµ ⊂ R(C). Now we shall show the opposite relation. Let ζ ∈ R(C) \ K. Then ζ ∈ R(A) \ K for some A ∈ C. The following cases are possible: either ζ = P (δ) for some δ ∈ A, or ζ = α(A), or ζ is zˆ corresponding to a couple of δ-corresponding points from < δ > for some δ ∈ A. In any case we have ζ ∈ Fν for some ν = 1, . . . , µ, and hence ζ ∈ Fµ . It means that in any case we have R(C) ⊂ Fµ and consequently R(C) = Fµ . And hence R(C) is simply connected and therefore schlichtartig (see [9, p. 175, 221-222]). We have proven that R(C) is an oriented, compact, simply connected, schlichtartig Riemann surface. Therefore from the known results (see [9, p. 175, 221-222; 5, p. 173]) it follows that R(C) is conformally equivalent to the Riemann sphere.

10

Proof of the main statement

Let Γ ∈ G. Using the the functions εΓ,j (ζ) (see Section 3), we shall define the following objects. For every edge γ of Γ let δγ+ and δγ− be the images of γ in the maps ζ 7→ εΓ,1 (ζ) and ζ 7→ εΓ,2 (ζ), respectively, and (δγ+ , δγ− ) =: δγ ∈ ∆1 . For every v ∈ W (Γ), let Av be the collection of all couples δγ corresponding to all edges γ incident to v. Then Av is a connectable star (see Section 5). Moreover, the collection {Av }v∈W (Γ) =: C(Γ)

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is a constellation which is acyclic, irreducible and connected (see Section 6). Let λΓ denote the conformal homeomorphism of the Riemann surface R(C(Γ)) onto ¯ C normalized by the conditions λΓ (0) = 0, λΓ (1) = ∞, λ′Γ (0) = 1. ¯ →C ¯ defined by the formula Consider the mapping φ : K φ = λΓ ◦ ηC(Γ) , and its inversion φ−1 . Let aj := εΓ,1(pj ), where pj runs over the whole set V (Γ) of nodes of order 1 of Γ, j = 1, . . . , n + 1. From the construction of R(C(Γ)) we see that the function φ is analytic at every ¯ holomorphic on K ¯ \ (W ∪ {1}), while its derivative φ′ is holomorphic and point of K, ¯ \ (W ∪ V ), has simple zeroes at all points of the not vanishing at all points of the set K set V \ {1} and the pole of order 3 at the point z = 1. For every v ∈ W (Γ) at all points β ∈ Wv , the function φ′ has an algebraic singularity such that the function (1/φ′)τ (v) is single-valued and holomorphic in a neighbourhood of every point β ∈ Wv , and at each such point it has a zero of order τ (v) − 2. Note that φ has no other critical points and ¯ Hence there is a positive integer l such that the function singularities on K. (1/φ′ )l ¯ (i.e in a neighbourhood of every point of K ¯ ). is single-valued and meromorphic on K For the constellation C := C(Γ), let us now consider the function f defined by the formula (3). From the above it follows that the function (f ′ /φ′ )l is single-valued, holo¯ But morphic and nowhere vanishing in K. (f ′ /φ′ )l = ((f ◦ φ−1 )′ )l and hence the right-hand side of this equality is single-valued, holomorphic and nowhere ¯ Therefore on C ¯ we have vanishing in C. ((f ◦ φ−1 )′ )l = const ∈ C \ {0} and hence each branch of the function (f ◦ φ−1 )′ is constant in C. So f ′ = cφ′ ¯ with c = const ∈ C \ {0}. But since on K f ′ (0) = cφ′ (0), |f ′ (0)| = 1

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and φ′ (0) = 1, then |c| = 1 and f (z) = cφ(z) ¯ in K. Let us show that the restriction fK of the function f = cφ to K is the extremal of the inner Tchebotar¨ov’s problem for the normalized point collection {aj } := {εΓ,1 (pj )} introduced in this section. Let g : K → C \ {aj } be any holomorphic univalent function with g(0) = 0. Then for every edge γ of Γ, the corresponding δγ := (δγ+ , δγ− ) and any couple (z + , z − ) of δγ -corresponding points (see Section 8), the closure of the union of f -images of radii of K ending at points z + and z − contains two non-intersecting arcs contained in g(K) and connecting the origin with the boundary of the set g(K). Hence in this situation all requirements of Theorems 1 and 2 from our work [12] are fulfilled and therefore |f ′(0)| ≥ |g ′(0)|, and the equality here is valid iff |g(z)| = |f (cz)| ∀z ∈ K with some constant c such that |c| = 1. It means that the function fK is the extremal of the inner Tchebotar¨ov’s problem for {aj } and that the extremal is unique up to rotation of K around the origin in the z-plane. From the above facts it is easy to derive all other statements of this work. 

References [1] G.M. Golusin: “Method of variations in the theory of conform representation”, I, Mat. Sb., Vol. 19(61), (1946), pp. 203–236, (in Russian). [2] G.M. Golusin:Geometric Theory of Functions of a complex Variable, AMS, Rhode Island, 1969 (transl. from Russian). ¨ [3] H. Gr¨otzsch : “Uber ein Variationsproblem der konformen Abbildung”, Ber. Ferh. S¨achs. Akad. Wiss. Leipzig, Vol. 82, (1930), pp. 251–263. [4] J.A. Jenkins: Univalent functions and conformal mappings, Inostr. Lit., Moscow, 1962, (Russian transl.). [5] S.L. Krushkal, B.N. Apanasov and B.N. Gusevskii: Kleinian groups and uniformisation in examples and problems, Nauka, Novosibirsk, 1981, (in Russian). [6] G.V. Kuz‘mina: “Covering theorems for functions holomorphic and univalent within a disk”, Dokl. Acad. Nauk SSSR, Vol. 160, (1965), pp. 25–28. (Russian.) [7] M.A. Lavrentieff” “Sur un probleme de maximum dans la representation conforme”, C. R. Acad. Sci. Paris, Vol. 191, (1930), pp. 827–829. [8] M.A. Lavrentyev : “On the theory of conformal mappings”, Trudy Fiz.-Mat. Inst. Akad. Nauk SSSR, Vol. 5, (1934), pp. 159–246, (in Russian).

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[9] R. Nevanlinna: Uniformisierung, Inostr. Lit., Moscow, 1955, (Russian transl.). [10] G. Polya: Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenh¨angende Gebiete, III, S.-B. Preuss. Akad. Wiss., Berlin, 1929, pp. 55– 62. [11] Ch. Pommerenke: Univalent Functions, Vandenhoeck and Ruprecht, G¨ottingen, 1975. [12] P.M. Tamrazov: “Theorems on the covering of lines under a conformal mapping”, Mat. Sb., Vol. 66(108), (1965), pp. 502–524. (Russian). [13] P.M. Tamrazov: Tchebotar¨ov’s extremal problem, Kiev, 2004; (Prepr. / National Acad. Sci. of Ukraine. Inst. of Math.; 2004.9). [14] P. Tamrazov: “Tchebotar¨ov’s problem”, C. R. Acad. Sci. Paris, Ser. I, (to be published).

CEJM 3(4) 2005 606–613

The Hurwitz determinants and the signatures of irreducible representations of simple real Lie algebras Alexander N. Rudy∗ Power Engineering Department, Belarus National Technical University, Minsk, Belarus

Received 26 April 2005; accepted 13 July 2005 Abstract: The paper deals with the real classical Lie algebras and their finite dimensional irreducible representations. Signature formulae for Hermitian forms invariant relative to these representations are considered. It is possible to associate with the irreducible representation a Hurwitz matrix of special kind. So the calculation of the signatures is reduced to the calculation of Hurwitz determinants. Hence it is possible to use the Routh algorithm for the calculation. c Central European Science Journals. All rights reserved.

Keywords: Lie algebras, signature formulae, Hurwitz matrix, Routh algorithm MSC (2000): 17B10, 17B20

1

Introduction

Consider a simple complex classical Lie algebra g and consider an irreducible representation ϕ : g → sl(V ). Denote by gσ the real form of inner type of g such that σ is a conjugation of algebra g with respect to gσ . Then ϕ(gσ ) ⊆ su(p, q). Let δ(gσ ) = p − q. Formulae for δ are derived in [1,4]. Suppose λ is the highest weight of the representation ϕ and suppose ρ is half the sum of the positive roots of g. Let R be a root system of algebra g. Denote by R+ the set of all positive roots of R and denote by R∨ the dual to R root system. If the rank r of g is small, then formulae for δ are rather simple [2-4] and are presented in terms of the highest weight coordinates. Formulae for arbitrary rank r P → → → algebras are derived in [1-4]. Let λ + ρ = h− ε , where − ε ,...,− ε is a standard q

q

1

r

q=1

basis of a realization space of the root system R [5, table I-IV]. If g = so(2r + 1, C), then [4] the calculation of δ is reduced to the calculation of determinants consisting of ∗

E-mail: [email protected]

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the coefficients bj of the polynomial expression r r Y X f (z) = (z − hj sin(πhj )) = bj (h)z r−j . j=1

(1)

j=0

It turns out rather unexpectedly that these determinants are the Hurwitz ones and so it is possible to apply to their calculation an appropriate technique [6, ch.XV].

2

Preliminaries

Let a1 , . . . , ak+m be any real numbers, k, m ∈ Z, k > 0, m > 0. For a polynomial expression k+m k+m Y X f (z) = (z − aj ) = bj z k+m−j . (2) j=0

j=1

denote by Hur(k + m) the (k + m) × (k + m) Hurwitz matrix 

b  1   b0  Hur(k + m) =  . . .  b2−(k+m)

b3 b2 .. .

. . . b2(k+m)−1 ... .. .

b4−(k+m) . . .



  b2(k+m)−2   , ..  .   bk+m

(3)

where bj = 0 for j < 0 and j > k + m. Denote by W (k, m) the determinant 1 a 1 ... k−2 a 1 W (k, m) = k−1 a1 k+1 a1 ... k+2m−1 a1

Lemma 2.1. W (k, m) = (−1)m Hm

Q

1

...

a2

...

...

...

ak−2 2

...

ak−1 2

...

ak+1 2

...

...

...

ak+2m−1 ... 2

(4)

(ai − aj ), where Hm is an order m upper

1≤j
left minor of the matrix Hur(k + m).

1 ak+m ... k−2 ak+m . ak−1 k+m k+1 ak+m ... ak+2m−1 k+m

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A.N. Rudy / Central European Journal of Mathematics 3(4) 2005 606–613

Proof. Consider the Vandermonde determinant Wk+2m

Wk+2m

1 a 1 ... k−1 a 1 = k a1 ... ak+2m−2 1 k+2m−1 a1

Wk+2m =

Y

... 1

1

...

. . . ak+m

z1

...

... ...

...

...

. . . ak−1 k+m

z1k−1

...

z1k

...

...

...

...

akk+m

... ...

. . . ak+2m−2 z1k+2m−2 . . . k+m . . . ak+2m−1 z1k+2m−1 . . . k+m

(zi − zj )

1≤j
m Y

f (zi )

i=1

Y

1 zm ... k−1 zm . k zm ... k+2m−2 zm k+2m−1 zm

(5)

(ai − aj ),

1≤j
where f (z) must be taken from (2). Furthermore,

Wk+2m

1 z1 = . . . m−1 z1

f (z1 ) z1 f (z1 ) = . . . m−1 z1 f (z1 )

1

...

z2 .. .

... .. .

z2m−1 . . .

f (z2 ) z2 f (z2 ) .. . z2m−1 f (z2 )

1 m Y zm Y · f (z (ai − aj ) i) .. . 1≤j
(6)

Making a Laplace decomposition of determinant (5) in its last m columns we find that k the summand z1k+2m−2 · z2k+2m−4 · . . . · zm has the coefficient (−1)((k+1)+(k+3)+...+(k+2m−1))+((k+m+1)+(k+m+2)+...+(k+2m)) W (k, m) = (−1)m(m+1)/2 W (k, m).

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On the other hand, from (6) it follows that the same summand has the coefficient b−m+2 b −m+3 ... b0 b1

b−m+4 . . . b−m+5 . . . ...

...

b2

...

b3

...

bm bm+1 Q (ai − aj ) . . . 1≤j
= (−1)[m/2] Hm

Q

(ai − aj ),

Y

(ai − aj ).

1≤j
where bj = 0 for j < 0 and j > k + m. Hence W (k, m) = (−1)m Hm

1≤j


3

The case g = so(2r + 1, C)

Let g = so(2r + 1, C). Consider a representation of g with the highest weight λ = (λ1 , . . . , λr ) : λ1 λ2 λr−1 λr d d ... d A d . (7)
That is λ = λ+ρ=

r P

q=1

r P

λi ωi , where ωi for i = 1, . . . , r are the basis representations of g. Let

i=1

→ hq − ε q , where

hq = 21 (2λq + 2λq+1 + . . . + 2λr−1 + λr + 2(r − q) + 1) for q = 1, 2, . . . , r − 1, hr = 12 (λr + 1). Let X(λ) = {β | β ∈ R+ ,

1 (β, (λ + ρ)) ∈ Z} 2

and let Cλ =

Y

β∈X(λ)

1 (β, (λ + ρ)). 2

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A.N. Rudy / Central European Journal of Mathematics 3(4) 2005 606–613

For any polynomial expression (1), denote by   b1 b3 . . .  b  0 b2 . . .  ..  .. .. . . . H(r) =    b4−r b6−r . . .   b  3−r b5−r . . .  b2−r b4−r . . .

H(r) the Hurwitz matrix  b2r−5 b2r−3 b2r−1   b2r−6 b2r−4 b2r−2    .. .. ..  . . .  ,  br−2 br br+2    br−3 br−1 br+1    br−4 br−2 br

(8)

where bj = 0 for j < 0 and j > r. Matrices like (8) where introduced by A.Hurwitz in 1896 and used for the study of stability of solutions of linear differential equations [6, ch.XV]. Consider the upper left minors of matrix (8):

b1 b3 H0 = 1, H1 = b1 , H2 = , . . . , Hr−2 b0 b2

b1 b0 = . . . b4−r

We associate the minor Hk with the algebra sor−k,r+k+1.

b3

...

b2 .. .

... .. .

b6−r . . .

b2r−5 b2r−6 . .. . br−2

Theorem 3.1. Let g = so(2r + 1, C) and gσ = sor−k,r+k+1 for k = 0, 1, . . . , r − 2. Let r P λ= λi ωi be the highest weight of a representation of g. i=1

If λr is odd, then δ = 0 for any gσ . If λr is even, then

|δ(sor−k,r+k+1)| =

2k(k+1)/2 |Hk |Cλ . (9) (0!1!2! · . . . · (r − k − 2)!)((r − k − 1)!(r − k + 1)! · . . . · (r + k − 1)!)

Proof. The proof partially coincides with the proof of Theorems 2 and 3 from [4]. If λr is odd, then δ = 0 for any gσ [4, theorem 2]. Let λr be even. Then [4, formula (9)]

|δ| = | lim t→1

2m(k) · 2r · det[cosap −1 (πthq )] ·

r Q

sin(πthj )

j=1

2r2 (π(t − 1))m(k) (0! · 2! · . . . · (2i − 2)!)(1! · 3! · . . . · (2r − 2i − 1)!)

|,

where m(k) = 12 (r 2 − r + k 2 + k) and i = (r − k)/2, if r − k is even, or i = (r + k + 1)/2, if r + k + 1 is even; ap = 2p − 1, if p = 1, . . . , i and ap = 2p − 2i, if p = i + 1, . . . , r, and det[cosap −1 (πthq )] denotes the determinant of an r × r matrix whose (p, q) element

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is cosap −1 (πthq ). We apply Lemma 2.1 to calculate this determinant. Then, leaving only the lowest degree of t − 1 in the numerator, we derive formula (9).  From formula (9) it follows that the calculation of δ is reduced to the calculation of the Hurwitz determinant Hk . We use Routh’s algorithm [6, ch.XV,§3] to triangulate matrix (8). This gives the easy way to calculate δ(gσ ) for the algebras sor−k,r+k+1 of arbitrary rank r. Corollary 3.2. Let λr be even. (i) Only the singular cases of first kind arise in the calculation of Hk by Routh’s algorithm, k = 0, 1, . . . , r − 3. (ii) Let δ(sor−k,r+k+1) = 0. Then λr is not necessarily odd. For the definition of singular cases of the first and second kind, see [6, ch.XV,§4]. Proof. Apply Routh’s algorithm to matrix (8). Namely, construct the table b0 b2 b4 . . . b2r−6 b2r−4 b2r−2 b1 b3 b5 . . . b2r−5 b2r−3 b2r−1 c2 c4 c6 . . . c2r−4 c2r−2 c2r

,

(10)

d3 d5 d7 . . . d2r−3 d2r−1 d2r+1 ... ... ... ... ...

...

...

where c2 = (b1 b2 − b0 b3 )/b1 c4 = (b1 b4 − b0 b5 )/b1 . . .

.

(11)

d3 = (c2 b3 − b1 c4 )/c2 d5 = (c2 b5 − b1 c6 )/c2 . . . Then [6, ch.XV, formula (32)] H1 = b1 , H2 = b1 c2 , H3 = b1 c2 d3 , . . .

.

Denote by det(H(r)) the determinant of matrix H(r) (8). Suppose that for some i = 1, . . . , r − 3 the i-th row of table (10) consists entirely of zeros. This is the singular case of the second kind. Then from formulae (11) follows that det(H(r)) = 0. But from Orlando’s formula [6, ch.XV, formula (40)] it follows that r(r+1)/2

det(H(r)) = (−1)

r Y Y (hj sin(πhj )) (hj sin(πhj ) + hi sin(πhi )). i=1

1≤j
And so det(H(r)) 6= 0. A contradiction proves (i). To prove (ii), consider the representation 0

1

d

d

0

d A


0 d

.

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A.N. Rudy / Central European Journal of Mathematics 3(4) 2005 606–613

Then

9 7 3 1 λ + ρ = ( , , , ), 2 2 2 2

and polynomial (1): f (z) = (z − 9/2)(z + 7/2)(z + 3/2)(z − 1/2) = z 4 −

189 35 2 z + 15z + 2 16

.

The Hurwitz matrix (8) is 

Construct table (10).

0

15

0

0



     1 −35/2 189/16 0    H(r) =  .   0 0 15 0     0 1 −35/2 189/16 1 −35/2 189/16 0 0

15

0

0

Since b1 = 0, it is the singular case of the first kind. Let b1 = ε. Then, using formulae (11) we obtain 1 −35/2 189/16 0 ε

15

0

0

−35/2 − 15/ε 189/16 0 15 +

189ε/16 −35/2−15/ε

0

189/16 Hence H0 = 1,

H1 = lim ε = 0, ε→0

H2 = lim ε(−35/2 − 15/ε) = −15. ε→0

So, using formula (9), we derive 1 · Cλ , (0! · 1! · 2!) · 3! 2 · 0 · Cλ |δ(so3,6 )| = = 0, (0! · 1!)(2! · 4!) 23 · 15 · Cλ |δ(so2,7 )| = . 1! · 3! · 5!

|δ(so4,5 )| =

From [4, table 2] it follows Cλ =

1 (λ1 + λ2 + λ3 + 3)(λ2 + 1)(λ2 + λ3 + λ4 + 3)(λ3 + λ4 + 2) 64 · (λ1 + 2λ2 + 2λ3 + λ4 + 6)(λ1 + λ2 + 2λ3 + λ4 + 5) = 48.

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Hence |δ(so4,5 )| =

48 = 4, 12

|δ(so3,6 )| = 0,

|δ(so2,7 )| =

8 · 15 · 48 = 8. 6 · 120



Acknowledgment The author is grateful to Professor B.P. Komrakov for setting the problem.

References [1] F.I. Karpelevich: “Simple subalgebras of real Lie algebras”, Trudy Mosk. Mat. Obshch., Vol. 4, (1955), pp. 3–112. [2] J. Patera and R.T. Sharp: “Signatures of finite su(p, q) representations”, J. Math. Phys., Vol. 25, (1984), pp. 2128–2131, MR0748387(85j:22042). [3] A.N. Rudy: “Signatures of finite representation of real, simple Lie algebras”, J. Phys. A:Math. Gen., Vol. 26, (1993), pp. 5873–5880, MR1252794(94i:17014). [4] A.N. Rudy: “Signatures of finite classical Lie algebra representations”, J. Phys. A:Math. Gen., Vol. 28, (1995), pp. 1641–1653, MR1338050(96e:17017). [5] N. Burbaki: Groupes et algebras de Lie. Ch. IV-VI, Hermann, Paris, 1968. [6] F.R. Gantmacher: The theory of matrices, AMS Chelsea Publishing, Providence, RI, 1959.

CEJM 3(4) 2005 614–626

Bivariant Chern classes for morphisms with nonsingular target varieties∗ Shoji Yokura† Department of Mathematics and Computer Science, Faculty of Science, University of Kagoshima, 21-35 Korimoto 1-chome, Kagoshima 890-0065, Japan

Received 31 March 2005; accepted 27 June 2005 Abstract: W. Fulton and R. MacPherson posed the problem of unique existence of a bivariant Chern class - a Grothendieck transformation from the bivariant theory F of constructible functions to the bivariant homology theory H. J.-P. Brasselet proved the existence of a bivariant Chern class in the category of embeddable analytic varieties with cellular morphisms. In general however, the problem of uniqueness is still unresolved. In this paper we show that e of for morphisms having nonsingular target varieties there exists another bivariant theory F e constructible functions and a unique bivariant Chern class γ : F → H. c Central European Science Journals. All rights reserved.

Keywords: Bivariant theory, Bivariant Chern class, Chern–Schwartz–MacPherson class, constructible function MSC (2000): 14C17, 14F99, 55N35

1

Introduction

W. Fulton and R. MacPherson [7] (also see [6]) introduced the notion of bivariant theory, which unifies both covariant and contravariant theories. They also defined transformations from one bivariant theory to another one, called Grothendieck transformations, which generalize ordinary natural transformations. In [7, §6 Whitney classes] they proved the unique existence of a bivariant Stiefel–Whitney class in the category of PL spaces and maps. Motivated by this, in [7, §10.4] they conjectured the existence of a bivariant Chern class, i.e., a Grothendieck transformation from the bivariant theory F of constructible ∗

Partially supported by Grant-in-Aid for Scientific Research (C) (No. 15540086 + No. 17540088), the Japanese Ministry of Education, Science, Sports and Culture. † E-mail: [email protected]

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functions to the bivariant homology theory H satisfying the property that for a morphism from a nonsingular variety X to a point the value of the characteristic function 11X of X is the Poincar´e dual of the total Chern cohomology class of X. If such a bivariant Chern class exists, then it follows that for morphisms to a point it becomes the Chern–Schwartz–MacPherson class transformation c∗ : F → H∗ (see [3], [11], [13], [15], [21], [22], etc.). This conjecture was solved by J.-P. Brasselet [1] in the category of embeddable complex analytic varieties whose morphisms are cellular. However, the question of whether “cellularness” of morphisms can be dropped and the problem of its uniqueness are still unresolved. In [16] C. Sabbah introduced another “micro-local” bivariant Chern class under certain restrictions, whereas the approach of J.-P. Brasselet [1] is based on methods of obstruction theory going back to the work of M.-H. Schwartz [21], [22]. In [27], [28] J. Zhou proved that for a morphism with a target variety being a nonsingular curve the bivariant Chern classes constructed by J.-P. Brasselet [1] and C. Sabbah [16] are both identical. We have generalized this result of Zhou’s to the following uniqueness theorem, in which the “cellularness” of morphisms is not assumed: Theorem 1.1. [25, Theorem (3.4)] If there exists a bivariant Chern class γ : F → H, then it is unique when restricted to morphisms whose target varieties are nonsingular and of any dimension; explicitly, for a morphism f : X → Y with Y nonsingular (of any f

dimension) and for any bivariant constructible function α ∈ F(X − → Y ) the bivariant Chern class γ(α) is expressed by f →Y) γ(α) = f ∗ s(T Y ) ∩ c∗ (α) ∈ H∗ (X) ∼ = H(X −

where s(T Y ) = c(T Y )−1 ∈ H ∗ (X) is the Segre cohomology class of the tangent bundle. f → Y ) for Y smooth is induced by multiplication The isomorphism H∗ (X) ∼ = H(X − f •[Y ] : H(X − → Y ) → H(X → pt) = H∗ (X) with the fundamental class [Y ] ∈ H(Y → pt) = H∗ (Y ) of Y , which, in down-to-earth terms, corresponds to the Alexander duality. This fact follows from the definition of the Fulton–MacPherson bivariant homology theory H, the definition of the bivariant product and the definition of the Alexander duality (e.g., see [6, Chapter 19]). Though it shall not be explicitly cited, this isomorphism is used in the following. This “twisted” Chern–Schwartz–MacPherson class f ∗ s(T Y ) ∩ c∗ (α) shall be called the Ginzburg–Chern class of a constructible function α (cf. [8], [9] and also [4]). Note that the target variety Y being nonsingular is essential for the definition of the Ginzburg–Chern class. Now it seems quite natural to ask the following question:

Question A: Does the correspondence defined by Ginzburg–Chern classes become a Grothendieck transformation ? To be more precise, for a morphism f : X → Y with Y being nonsingular we define f

f

γ Gin : F(X − → Y ) → H(X − →Y) by γ Gin (α) := f ∗ s(T Y ) ∩ c∗ (α).

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Since the Ginzburg–Chern class f ∗ s(T Y ) ∩ c∗ (α) is always defined for any arbitrary constructible function α ∈ F (X) as long as the target variety Y is smooth, the above question f

f

asks whether the transformation γ Gin : F(X − → Y ) → H(X − → Y ) is well-defined. More precisely, it asks whether on the Fulton–MacPherson’s bivariant theory F of constructible functions the Ginzburg–Chern class transformation γ Gin preserves the three basic bivariant operations; product, pushforward and pullback (for these three operations see §2). Our discussion leads us to the following question: e of constructible functions on which the Question B: Does there exist a bivariant theory F Ginzburg–Chern class transformation γ Gin preserves these three bivariant operations ?

We emphasize that we are considering a “bivariant theory of morphisms with nonsingular target varieties”, which is not a bivariant theory in the strict sense of [7]. Also, we do not necessarily consider some kinds of topological or geometric requirements on constructible functions such as the one used in the definition of the Fulton–MacPherson’s bivariant group F. Question B simply asks about the existence of a bivariant Chern class for morphisms e is defined. with nonsingular target varieties regardless of how the source bivariant group F e it follows that an affirmative By showing that there exists such a bivariant theory F, answer to Question A is equivalent to showing that the Fulton–MacPherson’s bivariant e Such an affirmative answer was recently group F is contained in the bivariant group F. announced in [2, Remark 5.4 (2)]. In this paper we will prove the following theorem, in which the “cellularness” of morphisms is not assumed: Theorem 1.2. For morphisms whose target varieties are nonsingular, there exists a e of constructible functions and furthermore the Ginzburg–Chern class bivariant theory F is the unique Grothendieck transformation e→H γ Gin : F

satisfying that γ Gin for morphisms to a point is the Chern–Schwartz–MacPherson class transformation c∗ : F → H∗ .

2

Bivariant Theories

For convenience of the reader we briefly recall some facts on the bivariant theory. Let C be a category with fiber products, final object pt and a class of “proper” maps, which is closed under composition and base change and which contains all identity maps. Furthermore, fiber square will be used as “independent square” in the sense of [7]. A bivariant theory B on a category C with values in an abelian category is an assignment to each morphism f X− →Y in the category C an abelian group f

B(X − →Y)

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which is equipped with the following three basic operations: (Product operations): For morphisms f : X → Y and g : Y → Z, the product operation f

f

gf

• : B(X − → Y ) ⊗ B(Y − → Z) → B(X −→ Z) is defined. (Pushforward operations): For morphisms f : X → Y and g : Y → Z with f proper, the pushforward operation gf g f⋆ : B(X −→ Z) → B(Y − → Z) is defined. (Pullback operations): For a fiber square g′

X ′ −−−→   f ′y g

X  f y

Y ′ −−−→ Y,

the pullback operation

f

f′

g ⋆ : B(X − → Y ) → B(X ′ − → Y ′) is defined. These three operations are required to satisfy the seven axioms (see [7, Part I, §2.2] for details): (B–1) product is associative, (B–2) pushforward is functorial, (B–3) pullback is functorial, (B–4) product and pushforward commute, (B–5) product and pullback commute, (B–6) pushforward and pullback commute, and (B–7) projection formula. B∗ (X) := B(X → pt) is a “module” theory and covariantly functorial for proper maps, id and B ∗ (X) := B(X − → X) is a “ring” theory and contravariantly functorial. Let B, B′ be two bivariant theories on a category C. Then a Grothendieck transformation from B to B′ γ : B → B′ is a collection of homomorphisms B(X → Y ) → B′ (X → Y ) for a morphism X → Y in the category C, which preserves the above three basic operations: (i) γ(α •B β) = γ(α) •B′ γ(β), (ii) γ(f⋆ α) = f⋆ γ(α), and (iii) γ(g ⋆ α) = g ⋆ γ(α).

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In the following we consider the category C of reduced complex spaces, which are assumed to be of finite dimension and of countable topology so that any such space has a topological embedding as a loced subset of some Euclidean space Rm . We take “proper” to have its usual topological meaning. The bivariant theories which we consider are bivariant theories of constructible functions and the Fulton–MacPherson bivariant homology theory. First, the abelian group F (X) of a given analytic variety X consists of all the constructible functions on X. The assignment X 7−→ F (X) becomes a contravariant functor with the usual pullback and at the same time covariantly functorial for proper holomorphic maps with the pushforward f∗ which takes the topological Euler–Poincar´e characteristic of the fibers weighted by constructible functions (see [15] and also [5], [10], [17], [20], [23]). The constructible function functor F itself can be a bivariant theory [26]: For any morphism f : X → Y the group sF(X → Y ) is defined by f

sF(X − → Y ) := F (X). This then becomes a bivariant theory with the following operations: (i) the product operation f

g

gf

• : sF(X − → Y ) ⊗ sF(Y − → Z) → sF(X −→ Z) is defined by α • β := α · f ∗ β, (ii) the pushforward operation gf

g

f⋆ : sF(X −→ Z) → sF(Y − → Z) is the usual pushforward f∗ : F (X) → F (Y ) and (iii) for a fiber square g′

X ′ −−−→   f ′y g

X  f y

Y ′ −−−→ Y,

f

(2.0)

f ′

the pullback operation g ⋆ : sB(X − → Y ) → sB(X ′ − → Y ′ ) is the functional pullback ∗ g ′ : F (X) → F (X ′ ). sF shall be called a simple bivariant theory of constructible functions. The axioms (B–2) and (B–3) clearly hold and one can prove that these three operations satisfy the other five axioms, using the following three properties: • for the fiber square (2.0) above, with g, g ′ proper,the following diagram commutes: f ′∗

F (Y ′ ) −−−→ F (X ′ )    g ′ g∗ y y∗ f∗

F (Y ) −−−→ F (X),

(2.1)

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• for a morphism f : X → Y and constructible functions α, β ∈ F (Y ) we have f ∗ (α · β) = f ∗ α · f ∗ β,

(2.2)

• (projection formula) for a proper morphism f : X → Y and constructible functions α ∈ F (Y ) and β ∈ F (X) we have f∗ (f ∗ α · β) = α · f∗ β.

(2.3)

One can show that with this simple bivariant theory sF there does not exist a Grothendieck transformation γ s : sF → H such that γ s (11π ) = c(T X) ∩ [X] for X smooth, where π : X → pt and 11π = 11X ([26]). f The Fulton–MacPherson bivariant group F(X − → Y ) of constructible functions consists of all the constructible functions on X which satisfy the local Euler condition with respect to f (see [1], [7], [16], [27], [28]). Here a constructible function α ∈ F (X) is said to satisfy the local Euler condition with respect to f if for any point x ∈ X and for any local embedding (X, x) → (CN , 0) the following equality holds
α(x) = χ Bǫ ∩ f −1 (z); α ,

which is the Euler–Poincar´e characteristic of Bǫ ∩ f −1 (z) weighted by the constructible function α, where Bǫ is a sufficiently small open ball of the origin 0 with radius ǫ and z is any point close to f (x) (cf. [1], [16]). The three operations on F are the same as above in sF and it is known that these three operations are well-defined for F. Note that idX F(X −−→ X) consists of all locally constant functions and F(X → pt) = F (X). The Fulton–MacPherson’s bivariant homology theory H is constructed from the usual cohomology theory with integer coefficients. For a morphism f : X → Y , choose a morphism φ : X → Rn such that Φ := (f, φ) : X → Y × Rn is a closed embedding (e.g., φ : X → Rn is already a closed embedding). Then the i-th bivariant homology group f

Hi (X − → Y ) is defined by

f

Hi (X − → Y ) := H i+n (Y × Rn , Y × Rn \ Xφ ), where Xφ is defined to be the image of the morphism Φ = (f, φ) (the definition is independent of the choice of φ). Note that instead of taking the Euclidean space Rn we can take any oriented manifold M so that i : X → M is a closed embedding and then consider the graph embedding f × i : X → Y × M. See [7, §3.1] for more details of H. In particular, note that if Y is nonsingular, H(X → Y ) is isomorphic to the homology group H∗ (X) of the source variety X by the Alexander duality isomorphism, as mentioned in the Introduction. Indeed, for a closed subspace X of an oriented real m-manifold M, capping with the fundamental class [M] of M induces the isomorphism ∼ = ∩[M] : H i (M, M \ X) − → Hm−i (X) (e.g., see [6, §19.1]).

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Main Theorem

In this section we give an affirmitive answer to Question B. f e Theorem 3.1. For a morphism f : X → Y with Y being nonsingular we define F(X − → Y ) to be the set of all constructible functions α ∈ F (X) satisfying the following two conditions (♯) and (♭) : for any fiber square g′

X ′ −−−→   f ′y g

X  f y

Y ′ −−−→ Y,

with Y ′ nonsingular (♯) the following equality holds for any constructible function β ′ ∈ F (Y ′ ): γ Gin (g ⋆ α • β ′ ) = γ Gin (g ⋆ α) • γ Gin (β ′ ) (i.e., c∗ (g ⋆ α • β ′ ) = γ Gin (g ⋆α) • c∗ (β ′)), (♭) γ Gin (g ⋆ α) = g ⋆ γ Gin (α). e becomes a bivariant theory with the same operations as in sF and furthermore Then F the Ginzburg–Chern class e→H γ Gin : F becomes the unique Grothendieck transformation satisfying that γ Gin for morphisms to a point is the Chern–Schwartz–MacPherson class transformation c∗ : F → H∗ . And also e F(X → pt) = F (X).

Proof. First we note that the uniqueness statement immediately follows from the proof given in [25], since it also applies to this “restricted” context of morphisms with nonsine satisfies the seven axioms, independent gular target varieties. Secondly, we note that F e becomes a bivariant theory, i.e., of conditions (♯) and (♭). Thus it suffices to show that F e For simplicity, we write γ that the three bivariant operations are well-defined on the F. Gin for γ . (i) the well-definedness of the pullback: Although it is more or less straightforward, for completeness, we provide a proof. Consider the following fiber squares: h′

g′

h

g

X ′′ −−−→ X ′ −−−→     ′′ ′ f y f y

X  f y

Y ′′ −−−→ Y ′ −−−→ Y.

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e Suppose that α ∈ F(X → Y ). Then we have

γ(h⋆ (g ⋆ α)) = γ((gh)⋆α)) (by (B–3)) = (gh)⋆ γ(α)

(since α satisfies (♭))

⋆ ⋆

= h g (γ(α)) (by (B–3)) = h⋆ γ(g ⋆ α)

(since α satisfies (♭) again).

Therefore g ⋆α satisfies condition (♭) with respect to the left fiber square. For condition (♯), we have to show that for any constructible function β ′′ ∈ F (Y ′′ ) we have γ(h⋆ (g ⋆ α) • β ′′ ) = γ(h⋆ (g ⋆ α)) • γ(β ′′ ). Indeed, γ(h⋆ (g ⋆α) • β ′′ ) = γ((gh)⋆ α • β ′′ )

(by (B–3))

= γ((gh)⋆ α) • γ(β ′′ ) ⋆

(since α satisfies (♯))

′′

⋆

= γ(h (g α)) • γ(β ) (by (B–3)). ′

f f e e ′− Hence α ∈ F(X − → Y ) implies g ⋆ α = g ′∗ α ∈ F(X → Y ′ ). f g e e (ii) the well-definedness of the product •: Let α ∈ F(X − → Y ) and β ∈ F(Y − → Z). We gf e need to show that α•β ∈ F(X −→ Z). Specifically, we need to show that for the following

fiber squares

h′′

X ′ −−−→   f ′y h′

Y ′ −−−→   g′ y h

X  f y Y  g y

Z ′ −−−→ Z if α and β satisfy the conditions (♯) and (♭) with respect to the top and bottom fiber squares, respectively, then the product α • β also satisfies (♯) and (♭) with respect to the outer fiber square. First we consider condition (♯): we need to show that for any constructible function δ ∈ F (Z ′ ) we have γ(h⋆ (α • β) • δ) = γ(h⋆ (α • β)) • γ(δ). Indeed, ⋆

γ(h⋆ (α • β) • δ) = γ(h′ α • h⋆ β • δ) ′⋆

⋆

′⋆

⋆

(by (B–3) and (B–5))

= γ(h α) • γ(h β • δ)

(since α satisfies (♯))

= γ(h α) • γ(h β) • γ(δ) (since β satisfies (♯)). e ′ → Y ′ ), thus by (♯) we get Now, (i) implies that h′ ⋆ α ∈ F(X ⋆

⋆

γ(h′ α • h⋆ β) = γ(h′ α) • γ(h⋆ β).

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Continuing with the above equations we get: ⋆

= γ(h′ α • h⋆ β) • γ(δ) (by (B–1))

= γ(h⋆ (α • β)) • γ(δ)

(by (B–5)).

Now we want to show that if α and β satisfy the conditions (♯) and (♭), then the product α • β also satisfies the condition (♭), i.e., γ(h⋆ (α • β)) = h⋆ γ(α • β). (Note that it is not clear at all whether α and β satisfying only the condition (♭) implies that the product α • β satisfies the condition (♭).) This can be seen as follows: ⋆

γ(h⋆ (α • β)) = γ(h′ α • h⋆ β)

(by (B–5))

′⋆

= γ(h α) • γ(h⋆ β) (since α satisfies (♯)) ⋆

= h′ γ(α) • h⋆ γ(β) (since α and β satisfy (♭))   = h⋆ γ(α) • γ(β) (by (B–5)) = h⋆ γ(α • β)

(since α satisfies (♯)).

f g gf e e − e Therefore, α ∈ F(X − → Y ) and β ∈ F(Y → Z) imply that α • β ∈ F(X −→ Z). (iii) the well-definedness of the pushforward: Consider the following fiber squares with f, f ′ being proper: h′′ X ′ −−−→ X    f f ′y y h′

Y ′ −−−→   g′ y h

Y  g y

Z ′ −−−→ Z.

gf e And let α ∈ F(X −→ Z). For condition (♭), we need to show that

γ(h⋆ f⋆ α) = h⋆ γ(f⋆ α)

and for condition (♯), we need to show that for any constructible function β ′ ∈ F (Z ′ ) we have γ(h⋆ f⋆ α • β ′ ) = γ(h⋆ f⋆ α) • γ(β ′ ).

Before we do this, we note that the homomorphism γ commutes with the pushforward for a proper morphism. This follows from the naturality of the Chern–Schwartz–MacPherson class transformation c∗ and the projection formula (for the cap product of cohomolgy class and homology class). We can now show that the above two formulas hold: γ(h⋆ f⋆ α) = γ(f ′ ⋆ h⋆ α) (by (B–6)) = f ′ ⋆ γ(h⋆ α) (since γ commutes with the pushforward) = f ′ ⋆ h⋆ γ(α) (since α satisifies (♭)) = h⋆ f⋆ γ(α)

(by (B–6))

= h⋆ γ(f⋆ α)

(since γ commutes with the pushforward).

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γ(h⋆ f⋆ α • β ′ ) = γ(f ′ ⋆ h⋆ α • β ′ ) ′

(by (B–6))

′

⋆

= γ(f ⋆ (h α • β )) ′

= f ⋆ γ(h α • β ) ′

(by (B–4))

′

⋆

623

(since γ commutes with the pushforward) ′

⋆

= f ⋆ (γ(h α) • γ(β )) (since α satisfies (♯) )

= f ′ ⋆ γ(h⋆ α) • γ(β ′ ) ′

(by (B–4))

′

⋆

= γ(f ⋆ h α) • γ(β ) ⋆

(since γ commutes with the pushforward)

′

= γ(h f⋆ α) • γ(β )

(by (B–6)).

e (iv) To prove that F(X → pt) = F (X), we need to show that for the fiber square p1

X × Y −−−→   p2 y g

Y

X  f y

−−−→ pt,

where Y is nonsingular, and for any constructible functions α ∈ F (X) and β ∈ F (Y ), c∗ (g ⋆α • β) = γ Gin (g ⋆ α) • c∗ (β)

(3.1.1)

γ Gin (g ⋆ α) = g ⋆ c∗ (α).

(3.1.2)

First we recall the following from [7, §2.4]: For morphisms f : X1 → X2 and g : Y1 → Y2 , the bivariant theoretic cross product × f

g

f ×g

× : B(X1 − → X2 ) ⊗ B(Y1 − → Y2 ) → B(X1 × Y1 −−→ X2 × Y2 ) is defined by α × β := p∗ α • q¯∗ β, where p : X2 × Y1 → X2 and q¯ : X2 × Y2 → Y2 are the projections. When X2 = Y2 = pt, or f and g are identiy morphisms, this gives us cross products on the covariant and contravariant groups. Thus, if the bivariant theory B is the bivariant homology theory H and when X2 = Y2 = pt, the above bivariant cross product is nothing but the homology cross product. Assuming that (3.1.2) holds, (3.1.1) becomes c∗ (g ⋆α • β) = g ⋆c∗ (α) • c∗ (β). It follows from our previous observations that g ⋆ c∗ (α) • c∗ (β) = c∗ (α) × c∗ (β) (homology cross product). For constructible functions ω ∈ F (W ) and ζ ∈ F (Z) the cross product ω × ζ ∈ F (W × Z) is defined to be (ω × ζ)(w, z) := ω(w)ζ(z). Then we have the following cross product formula of the Chern–Schwartz–MacPherson class ([12] and also see [14]), the proof of which uses the resolution of singularities: c∗ (ω × ζ) = c∗ (ω) × c∗ (ζ).

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By the definition we have c∗ (g ⋆ α • β) = c∗ (p∗1 α · p∗2 β)

= c∗ (α × 11Y ) · (11X × β) = c∗ (α × β)




= c∗ (α) × c∗ (β). Thus (3.1.1) holds under the assumption of (3.1.2). Therefore we only have to show that (3.1.2) holds. By the definition of γ Gin we have γ Gin (g ⋆α) = p∗2 s(T Y ) ∩ c∗ (p∗1 α)

= (1 × s(T Y )) ∩ c∗ (α × 11Y )   = (1 × s(T Y )) ∩ c∗ (α) × c∗ (11Y )   = (1 ∩ c∗ (α)) × s(T Y ) ∩ c∗ (Y ) 
 = c∗ (α) × s(T Y ) ∩ c(T Y ) ∩ [Y ] (since Y is nonsingular)  
= c∗ (α) × s(T Y ) ∪ c(T Y ) ∩ [Y ] = c∗ (α) × [Y ].

On the other hand, it follows from the definition of the bivariant pullback g ⋆ that g ⋆ c∗ (α) = c∗ (α) × [Y ]. Indeed, let i : X → M be a closed embedding into a manifold M and let AS,T : H ∗ (T, T \S) → H∗ (S) be the Alexander duality isomorphism defined by capping with the fundamental class [T ] for a closed subspace S of a manifold T (namely, AS,T (x) := x∩[T ]). If we use the cohomology pullback (idM ×g)∗ : H ∗ (M, M \ X) → H ∗ (M × Y, M × Y \ X × Y ), then g ⋆ (c∗ (α)) actually means the following: g ⋆ (c∗ (α)) = AX×Y,M ×Y ◦ (idM ×g)∗ ◦ A−1 X,M (c∗ (α)). Furthermore this is computed as follows: ⋆

g (c∗ (α)) = = = =

  −1 AX×Y,M ×Y ◦ (idM ×g) AX,M (c∗ (α))   −1 AX,M (c∗ (α)) × 1 ∩ [M × Y ]   −1 AX,M (c∗ (α)) × 1 ∩ ([M] × [Y ])   −1 AX,M (c∗ (α)) ∩ [M] × (1 ∩ [Y ])

= c∗ (α) × [Y ].

∗

S. Yokura / Central European Journal of Mathematics 3(4) 2005 614–626

Therefore (3.1.2) holds. This completes the proof.

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Example 3.2. For a smooth morphism f : X → Y of complex manifolds, the characterisf e tic function 11X on the source manifold X belongs to F(X − → Y ). And γ(11X ) = c(Tf )∩[X] with Tf being the relative tangent bundle of the morphism. The second property (♭) follows from g ′ ∗ Tf = Tf ′ and γ(11X )• corresponds to c(Tf ) ∩ f ∗ : H∗ (Y ) → H∗ (X) with f ∗ being the pullback of Borel–Moore homology. The first property (♯) follows from the so-called “Verdier–Riemann–Roch formula for Chern–Schwartz–MacPherson class” (see [7], [18] and [24]). Remark 3.3. J¨org Sch¨ urmann has recently generalized our construction of the bivariant e theory F to other theories, illustrating it for many examples (see [19]). Also see [2], in which the construction given in the present paper is extended in a suitable and more general way to arbitrary holomorphic maps.

Acknowledgment The author would like to thank J¨org Sch¨ urmann for his valuable comments and suggestions on an earlier version of the paper and also would like to thank the referee for his/her careful reading and useful comments.

References [1] J.-P. Brasselet: “Existence des classes de Chern en th´eorie bivariante”, Ast´erisque, Vol. 101–102, (1981), pp. 7–22. [2] J.-P. Brasselet, J. Sch¨ urmann and S. Yokura: “Bivariant Chern classes and Grothendieck transformations”, Math. AG/0404132. [3] J.-P. Brasselet and M.-H. Schwartz: “Sur les classes de Chern d’un ensemble analytique complexe”, Ast´erisque, Vol. 82–83, (1981), pp. 93–148. [4] N. Chriss and V. Ginzburg: Representation theory and complex geometry, Birkh¨auser, 1997. [5] A. Dimca: Sheaves in Topology, Springer-Verlag, 2004. [6] W. Fulton: Intersection Theory, Springer-Verlag, 1984. [7] W. Fulton and R. MacPherson: “Categorical frameworks for the study of singular spaces”, Memoirs of Amer. Math. Soc., Vol. 243, (1981). [8] V. Ginzburg: “G-Modules, Springer’s Representations and Bivariant Chern Classes”, Adv. in Maths., Vol. 61, (1986), pp. 1–48. [9] V. Ginzburg: “Geometric methods in the representation theory of Hecke algebras and quantum groups”, In: A. Broer and A. Daigneault (Eds.): Representation theories and algebraic geometry (Montreal, PQ, 1997), Kluwer Acad. Publ., Dordrecht, 1998, pp. 127–183. [10] M. Kashiwara and P. Schapira: Sheaves on Manifolds, Springer-Verlag, 1990.

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[11] G. Kennedy: “MacPherson’s Chern classes of singular algebraic varieties”, Comm. Algebra, Vol. 9(18), (1990), pp. 2821–2839. [12] M. Kwieci´ nski: “Formule du produit pour les classes caract´eristiques deChernSchwartz-MacPherson et homologie d’intersection”, C. R. Acad. Sci. Paris, Vol. 314, (1992) pp. 625–628. [13] M. Kwieci´ nski: “Sur le transform´e de Nash et la construction du graph de MacPherson”, In: Th`ese, Universit´e de Provence, 1994. [14] M. Kwieci´ nski and S. Yokura: “Product formula of the twisted MacPherson class”, Proc. Japan Acad., Vol. 68, (1992) pp. 167–171. [15] R. MacPherson: “ Chern classes for singular algebraic varieties”, Ann. of Math., Vol. 100, (1974), pp. 423–432. [16] C. Sabbah: Espaces conormaux bivariants, Th`ese, l’Universit´e Paris, Vol. 7, 1986. [17] P. Schapira: “Operations on constructible functions”, J. Pure Appl. Algebra, Vol. 72, (1991), pp. 83–93. [18] J. Sch¨ urmann: “A generalized Verdier-type Riemann–Roch theorem for Chern– Schwartz–Mac-Pherson classes”, math. AG/0202175. [19] J. Sch¨ urmann: “A general construction of partial Grothendieck transformations”, math. AG/0209299. [20] J. Sch¨ urmann: Topology of singular spaces and constructible sheaves, Monografie Matematyczne, Vol. 63, (New Series), Birkh¨auser, Basel, 2003. [21] M.-H. Schwartz: “Classes caract´eristiques d´efinies par une stratification d’unevari´et´e analytique complexe”, C. R. Acad. Sci. Paris, Vol. 260, (1965), pp. 3262–3264, 3535– 3537. [22] M.-H. Schwartz: “Classes et caract`eres de Chern des espaces lin´eaires”, Pub. Int. Univ. Lille, 2 Fasc. 3, (1980). [23] O. Viro: “Some integral calculus based on the Euler characteristic”, Springer Lect. Notes Math., Vol. 1346, (1989), pp. 127–138. [24] S. Yokura: “On a Verdier-type Riemann–Roch for Chern–Schwartz–MacPherson class”, Topology and Its Applications, Vol. 94, (1999), pp. 315–327. [25] S. Yokura: “On the uniqueness problem of the bivariant Chern classes”, Documenta Mathematica, Vol. 7, (2002), pp. 133–142. [26] S. Yokura: “Bivariant theories of constructible functions and Grothendieck transformations”, Topology and Its Applications, Vol. 123, (2002), pp. 283–296. [27] J. Zhou: Classes de Chern en th´eorie bivariante, Th`ese, Universit´e Aix-Marseille, Vol. 2, 1995. [28] J. Zhou: “Morphisme cellulaire et classes de Chern bivariantes”, Ann. Fac. Sci. Toulouse Math., Vol. 9, (2000), pp. 161–192.

CEJM 3(4) 2005 627–643

Self-adjoint differential vector-operators and matrix Hilbert spaces I∗ Maksim Sokolov† ICTP Affiliated Center, Department of Mechanics and Mathematics, The National University of Uzbekistan, Tashkent 700095, Uzbekistan

Received 20 March 2005; accepted 26 July 2005 Abstract: In the current work a generalization of the famous Weyl-Kodaira inversion formulas for the case of self-adjoint differential vector-operators is proved. A formula for spectral resolutions over an analytical defining set of solutions is discussed. The article is the first part of the planned two-part survey on the structural spectral theory of self-adjoint differential vector-operators in matrix Hilbert spaces. c Central European Science Journals. All rights reserved. ° Keywords: Self-adjoint differential vector-operators (direct sum operators), multi-interval quasidifferential systems, matrix spectral measure, Weyl-Kodaira inversion theorem MSC (2000): 34L05, 47B25, 47B37, 47A16

1

Introduction

The subject of study of this paper is a self-adjoint differential vector-operator (SADVO) which is one of two possible types of self-adjoint extensions of a minimal differential vector-operator on an Everitt-Markus-Zettl multi-interval system and which appears to be a direct sum operator comprising self-adjoint differential operators (coordinate operators). As was shown in a number of works [1-3], a SADVO is interesting for mathematical investigation from the point of view of structural spectral theory. These works deal with ordinary one-measure spectral theory, which shows that a SADVO inherits spectral properties of coordinate operators in a sophisticated way, contrary to the intuitive conjecture that a spectral property of a SADVO should be simply the direct sum of respective ∗ †

The work is dedicated to Professor Ravshan Ashurov on occasion of his 50-th anniversary. E-mail: [email protected]

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M. Sokolov / Central European Journal of Mathematics 3(4) 2005 627–643

spectral properties. Indeed, this can be seen quite easily. Let us have two self-adjoint differential operators T1 and T2 with simple spectra and corresponding ordered spectral representations U1 and U2 . In this case Fourier coefficients have the form Z (Ui fi )(λ) = fi (λ)θi (t, λ) dt. Ii

If the inheritance were trivial we would obtain U = U1 ⊕ U2 and thus Z 2 fi (λ)θi (t, λ) dt. (U (f1 ⊕ f2 ))(λ) = ⊕i=1 Ii

But the last equality is not correct in general, since a function of λ stands on the left hand side which is inconsistent with the vector-function of λ on the right side. This means that we need some indirect technique to build correct formulas of the mentioned type. So far we have already obtained the main one-measure results, but we need to study the spectral theory of SADVOs further. A theory of SADVO in matrix Hilbert spaces, in particular Weyl-Kodaira theorem and spectral resolutions formula, is important for several reasons. First of all, if we pass to matrix spectral measures, we will be able to construct inversion and resolution formulas based on analytical coordinate solution bases. If we obtain the analogs of Weyl-Kodaira theorems, it will be possible then to obtain an explicit formula for a matrix measure in terms of the solution bases and thus prove the analog of the important TitchmarshKodaira theorem. Secondly, it is also significant to show that a SADVO admits a theory which was previously known only for ordinary self-adjoint differential operators. That is, so far we have found objects which were not differential operators but for which we could prove theorems analogous to the respective theorems for ordinary differential operators. Let us proceed with a brief review of the basic notions of differential vector-operators and their spectral theory. Basic concepts of quasi-differential operators are well described in [4, 5]. Let Ω be a finite or a countable set of indices. On Ω, we have a multi-interval differential Everitt-Markus-Zettl system {Ii , τi }i∈Ω , where Ii are arbitrary intervals of the real line and τi are formally self-adjoint differential expressions of finite orders. This EMZ system generates a family of Hilbert spaces {L2 (Ii ) = L2i }i∈Ω and families of minimal {Tmin,i }i∈Ω and maximal {Tmax,i }i∈Ω differential operators. Consider the respective family {Ti }i∈Ω of self-adjoint extensions. Further, we introduce a system Hilbert space L(Ω) = ⊕i∈Ω L2i , consisting of vectors f = ⊕i∈Ω fi such that fi ∈ L2i and XZ X 2 2 |fi |2 dx < ∞. kfi ki = kf kL(Ω) = i∈Ω

i∈Ω

Ii

In the space L(Ω) consider the operator

T (Ω) : D(T (Ω)) ⊂ L(Ω) → L(Ω), defined on the dense domain D(T (Ω)) =

(

f ∈ L(Ω) :

X i∈Ω

)

kTi fi k2i < ∞

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by T (Ω)f = ⊕i∈Ω Ti fi . Definition 1.1. The operator T (Ω) is called a differential vector-operator generated by the self-adjoint extensions Ti , or a self-adjoint differential vector-operator. The operators Ti are called coordinate operators. The abstract preliminaries for this work may be found, for instance, in books [8, 9]. Fix i ∈ Ω. For each Ti there exists a unique resolution of the identity Eλi and a unitary operator Ui making the isometrically isomorphic mapping of the Hilbert space L2i onto the space L2 (Mi , µi ) where the operator Ti is represented as a multiplication operator. Below we review the structure of the mapping Ui . A vector φ ∈ L2i is called a cyclic vector if for each z ∈ L2i there exists a Borel function f such that z = f (Ti )φ. Generally there is not one cyclic vector in L2i but a collection {φk } such that L2i = ⊕k L2i (φk ), where L2i (φk ) are Ti -invariant subspaces in L2i generated by the cyclic vectors φk . That is L2i (φk ) = {f (Ti )φk }, for a varying Borel function f , such that φk ∈ D(f (Ti )). A vector φ ∈ L2i is called maximal relative to the operator Ti if each measure (E i (·)x, x)i , x ∈ L2i , is absolutely continuous with respect to the measure (E i (·)φ, φ)i . For each Hilbert space L2i there exists a unique (up to unitary equivalence) decomposition L2i = ⊕k L2i (ϕki ), where ϕ1i is maximal in L2i relative to Ti and a decreasing set of multiplicity sets eik , where ei1 is the whole line, such that ⊕k L2i (ϕki ) is equivalent with ⊕k L2 (eik , µi ), where the measure of the ordered representation is defined as µi (·) = (E i (·)ϕ1i , ϕ1i )i . A spectral representation of Ti in ⊕k L2 (eik , µi ) is called the ordered representation and it is unique, up to a unitary equivalence. Two operators are called equivalent, if they create the same ordered representation of their spaces. A well-known theorem [9, Ch. XIII, Section 5, Theorem 1]) represents the structural result for the ordered representation of the operator Ti in its abstract form. Since the generalized eigenfunctions Wk (x, λ) from this theorem are only measurable with respect to the spectral parameter λ, the usual technique is to decompose them using an analytical basis of solutions of the equation (τi − λ)σ = 0. At that, frequently we do not need all the basis functions and use only some of them. A Defining system σ1 , . . . , σs is the subsystem of the solution basis such that all Wk (·, λ) belong to its linear capsule. This treatment leads to an important conception of matrix Hilbert spaces. These are defined as follows. Let ∆ be an open interval and let Rn be a positive (n × n)-matrix measure on ∆, that is (I) Rn (e) is an (n × n)-matrix {̺pq (e)}, which consists of complex-valued set functions, hermitian and positively semidefinite for each Borel set e having a compact closure in ∆; (II) for each sequence of non-intersecting subsets in ∆ with a precompact union in ∆,

Rn (∪∞ m=1 em ) =

∞ X

m=1

Rn (em ).

Let us have a matrix measure Rn with components which are absolutely continuous

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relative to a measure of an ordered representation µ. The formula Z ̺pq (∆) = mpq (λ)dµ ∆

defines an (n × n)-density matrix {mpq }. The sets F = {f1 , f2 , . . . , fn } of Borel functions defined on ∆, for which ) Z (X n mpq (λ)fp (λ)fq (λ) dµ(λ) < ∞, kF k2Rn = ∆

p,q=1

form the space L20 (∆, Rn ). If kF kRn = 0, then F is called Rn -zero function. A matrix Hilbert space L2 (∆, Rn ) is defined as equivalence classes of elements from L20 (∆, Rn ) modulo Rn -zero functions. Definition 1.2. For i ∈ Ω, we introduce also a sliced union of sets Mi as a set M , containing all Mi on different copies of ∪i∈Ω Mi . The sets Mi do not intersect in M , but they can superpose, i.e. two sets Mi and Mj superpose, if their projections in the set ∪i∈Ω Mi intersect. For zi ∈ L2i , i ∈ Ω, define zbi = {0, ..., 0, zi , 0, ..., 0} ∈ L(Ω), where zi is on the i-th place. Definition 1.3. For each i ∈ Ω, let δ(Ti ) denote the subspectrum of the operator Ti , i.e. δ(Ti ) = σpp (Ti ) ∪ σcont (Ti ). Note that δ(Ti ) = σ(Ti ), since σpp (Ti ) equals to the set of eigenvalues and may not be closed. For instance, the subspectrum of an operator having the complete system of eigenfunctions with eigenvalues being the rational numbers of [0, 1] equals to Q ∩ [0, 1]; the subspectrum of an operator having the continuous spectrum [0,1] is assumed to equal to [0,1] without loss of generality. For this see also [8, Chapter VII.2]. Keep in mind also a projecting mapping P : M → ∪i∈Ω Mi such that P (δ(Ti )) = δ(Ti ). Definition 1.4. Let Ω = ∪K k=1 Υk , Υk ∩ Υs = ∅ for k 6= s and Υk = {s ∈ Ω : ∀s, l ∈ Υk , s 6= l, P (δ(Ts )) ∩ P (δ(Tl )) = Bsl ,

where kEBt sl ϕt k2t = 0 for at least one cyclic ϕt ∈ L2t , t = s, l}.

From all such divisions of Ω we choose and fix the one, which contains the minimal number of Υk . In the case when all the coordinate spectra σ(Ti ) are simple, we define the number Λ = Λ(T (Ω)) = min{K} as the spectral index of the vector-operator T (Ω).

1.1 List of one-measure theory results Here we list all the necessary theorems, proved in [1], [2] and [3] with some comments.

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Theorem 1.5. [1, 2] The identity resolution {Eλ } of the vector-operator T (Ω) equals to the direct sum of the coordinate identity resolutions {Eλi }, that is {Eλ } = ⊕i∈Ω {Eλi } This is a general theorem proved for abstract SAVOs. Here the inheritance mentioned in the beginning of the introduction is still trivial. Theorem 1.6. [1, 2] Let each Ti have a cyclic vector ai in L2i . Then the vector-operator P T (Ω) has at least Λ cyclic vectors {ak }Λk=1 , having the form ak = i∈Υk abi .

This is also a general theorem but the inheritance is not trivial now since Theorem 1.6 states that if we take direct sums of coordinate cyclic vectors we will not obtain a single cyclic vector in general but a collection of them. Moreover, in some cases (distorted vector-operators [2]) it is not possible to obtain the minimal possible number of cyclic vectors using Theorem 1.6. That is, Theorem 1.6 needed to be strengthened and such strengthening was contained in so called method of division on subspectra (MDS) leading to the construction of an ordered spectral representation. This method was developed in [2]. Briefly, in applying MDS you cut coordinate cyclic vectors on non-zero-measure sets of their superposition and only then unite the obtained pieces of cyclic vectors in direct sums. Combinatorially one can then obtain the minimal possible number of cyclic vectors (see Appendix). Let I = ∨i∈Ω Ii denote the sliced union of intervals Ii . Similarly, I k = ∨j∈Υk Ij . If xi are variables on Ii , then ∨xi will designate a variable either on I or I k depending on the context. This notation shows that a vector-function z = {z1 (x1 ), . . . , zn (xn ), . . . } on I or I k may be written as z(∨xi ). In particular, we may also write z(∨xi ) instead of z = ⊕i∈Ω zi . Let us introduce the space ⊕i∈Ω L∞ (Iin ). Here, z(∨xi ) ∈ ⊕i∈Ω L∞ (Iin ) means that ) ( sup ess sup |zi (xi )| xi ∈Iin

i∈Ω

< ∞,

∞ n where for each i, families {Iin }∞ n=1 represent compact subintervals of Ii , such that ∪n=1 Ii = Ii . In [6, Lemma 2.1], it was shown that ⊕i∈Ω L∞ (Iin ) = (⊕i∈Ω L1 (Iin ))∗ , where the space of Lebesgue-integrable vector-functions ⊕i∈Ω L1 (Iin ) is defined analogously to L(Ω). R We also need to introduce a symbolic integral ∨Ji f (∨xi ) d(∨xi ) defined by: ¾ ½Z Z Z f2 (x2 ) dx2 , · · · , f1 (x1 ) dx1 , f (∨xi ) d(∨xi ) = ∨Ji

J1

J2

where f (∨xi ) is understood to be measurable relative to d(∨xi ) if fi (xi ) are measurable relative to Lebesgue measures dxi . Then Z f (∨xi ) d(∨xi ) < ∞ ∨Ji

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if supi

R

Ji

|fi (xi )|dxi < ∞.

Theorem 1.7 ([3]). Let T (Ω) be a SADVO, generated by an EMZ system {Ii , τi }i∈Ω . Let U be an ordered representation of the space L(Ω) = ⊕i∈Ω L2 (Ii ) relative to T (Ω) with the measure θ and the multiplicity sets sk , k = 1, m. Then there exist kernels Θk (∨xi , λ), measurable relative to d(∨xi ) × θ, such that Θk (∨xi , λ) = 0 for λ ∈ R \ sk and (⊕i∈Ω τi − λ)Θk (∨xi , λ) = 0 for each fixed λ. Moreover for any bounded Borel set ∆, Z

∆

|Θk (∨xi , λ)|2 dθ(λ) ∈ ⊕i∈Ω L∞ (Iin ) ∀n ∈ N.

k

(U w) (λ) = lim

n→∞

Z

In

w(∨xi ) Θk (∨xi , λ) d(∨xi ), w ∈ L(Ω),

(1)

(2)

where the limit exists in L2 (sk , θ). The kernels {Θk (∨xi , λ)}nk=1 , n 6 m, are linearly independent as vector-functions of the first variable almost everywhere relative to the measure θ on sn . This is a structural theorem involving the differential structure of the coordinate operators. The proof is also based on the MDS, but it is much more combinatorially complicated than the MDS itself. In the latter we had to find only one maximal cyclic vector and did not need the other, since they could be obtained through the maximal measure applied on respective multiplicity sets. But in Theorem 1.7 we have to find all eigenfunctions, not only the ones corresponding to the maximal cyclic vector. Note that on the right hand side of (2) is a direct sum of functions of λ which is here equivalent to a regular sum of the functions, which is in accordance with the left hand side. Theorem 1.8 (Eigenfunction expansions, [3]). For any w ∈ L(Ω), there exists a decomposition Z +n m X (U w)k (λ)Θk (∨xi , λ) dθ(λ), lim w= k=1

n→∞

−n

This is just a corollary of Theorem 1.7. Theorem 1.9 ([3]). Let T (Ω) be a SADVO, generated by an EMZ system {Ii , τi }i∈Ω . Let the measure θ and the sets {sk }m k=1 be respectively a measure and multiplicity sets of an ordered representation of the space L(Ω) = ⊕i∈Ω L2 (Ii ), relative to the operator T (Ω). The kernels {Θk }m k=1 are the generalized vector-operator eigenfunctions, corresponding to the multiplicity sets (as defined in Theorem 1.7). Given a bounded Borel function F , which equals zero beyond a bounded Borel set ∆, the bounded vector-operator F (T (Ω)) may be represented as an integral operator: [F (T (Ω))f ](∨si ) = lim

n→∞

Z

In

f (∨xi )K(F ; ∨si , ∨xi ) d(∨xi ),

(3)

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633

where f ∈ L(Ω) and K(F ; ∨si , ∨xi ) =

m Z X k=1

F (λ)Θk (∨xi , λ)Θk (∨si , λ) dθ(λ).

(4)

∆

Since the kernels from Theorem 1.7 are only measurable relative to λ, the following theorem is important to strengthen the practical value of theorems 1.7, 1.8 and 1.9: Theorem 1.10 ([3]). Each kernel Θk (∨xi , λ), k = 1, m, may be decomposed as Θk (∨xi , λ) =

Mk X

γsk (λ)σsk (∨xi , λ),

(5)

s=1

where the Mk are finite for each k and σsk (∨xi , λ) depend analytically on λ as vectorfunctions. In this theorem the kernels σsk depend on k since they are specially constructed for the decomposition of Θk . This circumstance is inherent in the theory of differential vectoroperators and is one of the principal moments which make this theory different from that of ordinary differential operators. So if we want to deal with λ-analytical functions, we have to develop the matrix Hilbert space spectral theory. At the end of the paper [3], we have obtained the matrix spectral measure. Below we briefly review how we did this. In Theorem 1.10 we have obtained the formula (5) which we substitute now in (9). Thus we obtain: Mk m Z X X K(F ; ∨si , ∨xi ) = F (λ)γsk (λ)γpk (λ)σsk (∨si , λ)σpk (∨xi , λ) dθ(λ). k=1

∆ s,p=1

The last formula may be rewritten as ) Z (X Mk K(F ; ∨si , ∨xi ) = F (λ)σsk (∨si , λ)σpk (∨xi , λ) d̺sp (λ) , ∆

s,p=1

where ̺sp (∆) =

m Z X k=1

γsk (λ)γpk (λ) dθ(λ).

∆

Separate arguments show that RMk (·) = {̺sp (·)} is a correctly constructed matrix measure.

2

Weyl-Kodaira inversion theorem

The main result of the current work is the theorem presented in this section. This is the generalization of well-known Weyl-Kodaira inversion theorems ([9, Theorems XIII.5.13 and XIII.5.14]).

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Theorem 2.1. Let T (Ω) be a SADVO, generated by an EMZ system {Ii , τi }i∈Ω . Let ∆ be nj an open real interval and let {σj (·, λ)}j=1 be the defining systems of solutions of τi σ = λσ on I × ∆ with corresponding vector-functions σsk (∨xi , λ) obtained in Theorem 1.10. Then 1. There exists a positive (M × M )-matrix measure RM , M = max{nj }, defined on ∆, such that a limit

{(V f )s (λ)}M s=1 =

(

Z

lim

n→∞

)M

f (∨xi )σsk (∨xi , λ) d(∨xi )

∨i∈Ω Iin

s=1

exists in the topology of L2 (∆, RM ) for each f ∈ L(Ω) and defines an isometric isomorphism of E∆ L(Ω) onto L2 (∆, RM ). {(V f )s (λ)}M s=1 depends implicitly on k ∈ {1, ..., m}. −1 2. The inversion V is given by

(V −1 F )(∨xi ) = lim

∆n →∆

Z

∆n

(

M X

)

Fs (λ)σpk (∨xi , λ) d̺sp (λ) ,

s,p=1

where F = {F1 , ..., FM } ∈ L2 (∆, RM ) and the limit exists in L(Ω) for a set {∆n } of open subintervals in ∆ with compact closures. At that (V −1 F )(∨xi ) depends implicitly on k ∈ {1, ..., m}. Proof. Throughout the proof, keep in mind that m may equal infinity. 1. The density of the measure ̺sp relative to θ is the function

msp (λ) =

m X

γsk (λ)γpk (λ).

k=1

Let M = max{Mk } for γsk (λ) = 0, s > Mk . For each set of Borel functions F = {f1 , f2 , . . . , fM } defined on ∆, define a mapping Γ:

ΓF = {g1 , g2 , . . . , gm }, gk (λ) =

M X

γsk (λ)fs (λ).

s=1

Note that γsk (λ) = 0 when λ 6∈ sk (here sk is the k-th multiplicity set of the operator

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T (Ω)). Since kF k2Rn = =

Z (X M ∆

)

msp (λ)fs (λ)fp (λ)

s,p=1

Z (X m X M

dθ(λ) =

γpk (λ)fp (λ)γsk (λ)fs (λ)

∆

)

dθ(λ) =

k=1 s,p=1   Z X m M M  X X  = γpk (λ)fp (λ) γsk (λ)fs (λ) dθ(λ) =  ∆  k=1 p=1 s=1 ) Z (X m m Z X 2 = |gk (λ)| |gk (λ)|2 dθ(λ) = dθ(λ) = ∆

=

m X k=1

k=1

k=1

∆

kgk k2k = kΓF k2 ,

the mapping 2 m 2 Γ : L2 (∆, RM ) → ⊕m i=1 L (si ∩ ∆, θ) ⊆ ⊕i=1 L (si , θ)

is an isomorphic isomorphism. Let U be the isomorphic isomorphism from Theorem 1.7. From Theorem 1.7 it follows that if {Iin } are increasing sets of intervals, such that ∪n Iin = Ii , then for each n and for each f ∈ L(Ω), Z f (∨xi )Θk (∨xi , λ) d(∨xi ) ∈ L2 (sk , θ), k = 1, m. ∨i Iin

Moreover, lim

n→∞

Z

f (∨xi )Θk (∨xi , λ) d(∨xi )

∨i∈Ω Iin

=

Z

f (∨xi )Θk (∨xi , λ) d(∨xi )

I

exists in the topology of the space L2 (sk , θ). The mapping Z f → χ∆ (λ) f (∨xi )Θk (∨xi , λ) d(∨xi ), f ∈ E∆ L(Ω) I

is an isometric isomorphism of the space E∆ L(Ω) onto 2 ⊕m i=1 L (si ∩ ∆, θ).

For a function f ∈ L(Ω), define (Vn f )s (λ) =

Z

f (∨xi )σsk (∨xi , λ) d(∨xi ),

∨i∈Ω Iin

m Γ{(Vn f )s (λ)}M s=1 = {gk (λ)}k=1 ,

=

(U f)k (λ)

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where σsk have the structure of the direct sum, i (xi , ·) σsk (∨xi , ·) = ⊕i∈Ω σsk i and σsk (xi , λ) are specially chosen pieces of functions from the solution bases obtained by applying the MDS (see Appendix 2). Note that (Vn f )s (λ) depends implicitly on k: it itself consists of m components. Using Theorem 1.10, we can obtain the equality

(ΓVn f )k (λ) = =

M X

Zs=1

γsk (λ)

Z

f (∨xi )σsk (∨xi , λ) d(∨xi ) =

∨i Ini

f (∨xi )Θk (∨xi , λ) d(∨xi ).

∨i Ini

So we derive

2 k{(Vn f )s }M s=1 kRM =

=

Z (X M ∆

)

msp (λ)(Vn f )s (λ)(Vn f )p (λ)

s,p=1

ÃM Z (X m X ∆

k=1

γpk (λ)(Vn f )p (λ)

M X

dθ(λ) =

γsk (λ)(Vn f )s (λ)

s=1

p=1

!)

dθ(λ) =

   Z X m M M  X X   = γpk (λ)(Vn f )p (λ) γsk (λ)(Vn f )s (λ) dθ(λ) =  ∆  k=1 p=1 s=1  ¯M ¯2  Z X m ¯X ¯ ¯ ¯ γpk (λ)(Vn f )p (λ)¯ = dθ(λ) = ¯ ¯ ∆  k=1 ¯ p=1 m Z X |(ΓVn f )k (λ)|2 dθ(λ) < ∞, = k=1

∆

2 which means that {(Vn f )s }M s=1 ∈ L (∆, RM ). Moreover, M {(V f )s (λ)}M s=1 = { lim (Vn f )s (λ)}s=1 n→∞

exists in the topology of L2 (∆, RM ). From all of the above it follows that U E∆ = ΓV and V is an isometric isomorphism from E∆ L(Ω) onto L2 (∆, RM ). 2. Now prove the second part of the theorem. Let f ∈ L(Ω) be such that (V f )p (λ) are bounded for p = 1, M , and let g = ⊕i∈Ω gi be a vector-function whose coordinate elements equal zero beyond a set of compact intervals

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637

{Ji }i∈Ω . Let ∆0 be an open interval in ∆ with a compact closure. Then (E∆0 f , g)L(Ω) = (E∆0 f , E∆0 g)L(Ω) = (V E∆0 f , V E∆0 g)RM = ) Z (X M (V f )s (λ)(V g)p (λ) d̺sp (λ) = = ∆0

Z

=

∆0

Z

=

s,p=1

(

M X

(V f )s (λ)

∨i∈Ω Ji

s,p=1

∨i∈Ω Ji

µZ

(Z

M X

) ¶ g(∨xi )σpk (∨xi , λ) d(∨xi ) d̺sp (λ) = )

[(V f )s (λ)σpk (∨xi , λ) d̺sp (λ)] g(∨xi ) d(∨xi ).

∆0 s,p=1

(6)

For any e f ∈ L(Ω), let {fk } be a sequence in L(Ω), such that corresponding to each e fk there is a set {(V fk )p (λ)}M p=1 with bounded elements. fk → f implies (V fk )p (λ) → (V e f )p (λ) for each p = 1, M . That is Z

M X

∆0 s,p=1

(V fj )s (λ)σpk (∨xi , λ) d̺sp (λ) →

Z

M X

∆0 s,p=1

(V e f )s (λ)σpk (∨xi , λ) d̺sp (λ),

for each ∨xi ∈ ∨i∈Ω Ji . Applying formula (6) to each fk and then taking the limit, we derive ) (Z Z M X [(V e f )s (λ)σpk (∨xi , λ) d̺sp (λ)] g(∨xi ) d(∨xi ). (7) (E∆0e f , g)L(Ω) = ∨i∈Ω Ji

∆0 s,p=1

Since (7) is valid for any g ∈ L(Ω), we have the formula

from which follows

E∆0e f=

Z

∆0

E∆e f = lim

∆0 →∆

and the proof of the theorem.

(

M X

s,p=1

Z

∆0

(

)

(V e f )s (λ)σpk (∨xi , λ) d̺sp (λ) ,

M X

s,p=1

(V e f )s (λ)σpk (∨xi , λ) d̺sp (λ)

)

The next theorem can be easily obtained; we give it without a proof. Theorem 2.2 (Spectral resolutions). Let T (Ω) be a SADVO, generated by an EMZ nj be the defining system {Ii , τi }i∈Ω . Let ∆ be an open real interval and let {σj (·, λ)}j=1 systems of solutions of τi σ = λσ on I × ∆ with corresponding vector-functions σsk (∨xi , λ) obtained in Theorem 1.10. Let the matrix spectral measure RM be taken from Theorem 2.1. Then, given a bounded Borel function F , which equals zero beyond a Borel set Π,

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precompact in ∆, the bounded vector-operator F (T (Ω)) may be represented as an integral operator: Z [F (T (Ω))f ](∨si ) = lim f (∨xi )K(F ; ∨si , ∨xi ) d(∨xi ), (8) n→∞

where f ∈ L(Ω) and

K(F ; ∨si , ∨xi ) =

3

Z (X M Π

In

)

F (λ)σsk (∨si , λ)σpk (∨xi , λ) d̺sp (λ) .

p,s=1

(9)

Conclusion

We have obtained the necessary results for a deeper investigation of the matrix spectral measure. In the future, we plan to present an explicit formula for the matrix measure and thus prove a Titchmarsh-Kodaira theorem.

A

Appendix

In order to make the article self-contained we describe here some important ideas about the structural spectral theory of SADVOs.

A.1 The method of division on subspectra Since the MDS appeared to be the main tool for proving the theorems discussed above, it is reasonable to present its structure here. For a complete discussion, consult [2]. We demonstrate the MDS for finding a maximal vector in the direct sum space. Let ai be maximal vectors relative to the operators Ti in L2i . We want to find a maximal vector relative to the vector-operator T (Ω). We know that the vector ⊕i∈Ω ai does not give a single measure if a set P (δ(Ti ))∩P (δ(Tj )) has a non-zero spectral measure for i 6= j. Consider restrictions Ti |L2i (ai ) = Ti′ . Since all the operators Ti′ have single cyclic vectors ai , we can divide Ω into Υk , k = 1, Λ and apply Theorem 1.6 for the operator ⊕i∈Ω Ti′ . Then we derive Λ vectors ak = ⊕j∈Υk aj which are maximal in the respective spaces L(Υk ) = ⊕j∈Υk L2j (aj ). Now let 1 < Λ < ∞. Define T (Υk ) = ⊕j∈Υk Tj′ . For any two operators T (Υk ) and T (Υs ), k 6= s, let us introduce the sets δk,s = P (δ(T (Υk ))) ∩ P (δ(T (Υs ))) and δk = P (δ(T (Υk ))) \ δk,s . There exist unitary operators U k : L(Υk ) → L2 (R, µak ). Consider measures µk and µk,s , defined as µk,s (e) = µak (e ∩ δk,s ) and µk (e) = µak (e ∩ δk ), for any measurable set e. For any operator T (Υk ) (with respect to T (Υs )), measures µk and µk,s are mutually singular and µk + µk,s = µak ; therefore L2 (R, µak ) = L2 (R, µk ) ⊕ L2 (R, µk,s ).

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This means that (according to our designations): Uk

−1

: L2 (R, µak ) −→ L(akk ) ⊕ L(akk,s )

and ak = akk ⊕ akk,s ,

(10)

where akk and akk,s form the measures µk and µk,s respectively. Let max{w, ψ} denote the vector which is maximal of the two vectors in the brackets (Note that this designation is valid only for vectors, considered on the same set. In order not to complicate the investigation we assume here that any two vectors are comparable in this sense. In order to achieve this, it is enough to decompose each coordinate operator Ti into the direct sum Tipp ⊕ Ticont , where the operators have respectively pure point and continuous spectra. Then after redesignation we obtain the equivalent vector-operator to the initial vectoroperator ⊕Ti ). Consider first two operators T (Υ1 ) and T (Υ2 ). It is clear that the vector © ª a1⊕2 = a11 ⊕ a22 ⊕ max a11,2 , a22,1

is maximal in L(Υ1 ) ⊕ L(Υ2 ). Note that both a11 and a22 may equal zero. The maximal vector in L(Υ1 ) ⊕ L(Υ2 ) ⊕ L(Υ3 ) will have the form: © 1⊕2 ª 3 3 a1⊕2⊕3 = a1⊕2 1⊕2 ⊕ a3 ⊕ max a1⊕2,3 , a3,1⊕2 ,

1⊕2 where a1⊕2 , corresponding to the set which is free from the 1⊕2 is the narrowed vector a superposition with δ(T3 ), as shown in (10). Continuing this process, we obtain a maximal vector in the main space L(Ω):

ª © 1⊕···⊕Λ−1 1⊕···⊕Λ−1 , aΛΛ,1⊕···⊕Λ−1 . a1⊕···⊕Λ = a1⊕···⊕Λ−1 ⊕ aΛΛ ⊕ max a1⊕···⊕Λ−1,Λ

(11)

° ° ° ° °[⊕i∈Ω E i (·)] a1⊕···⊕Λ °2 = lim °[⊕Lj=1 E j (·)] a1⊕···⊕L °2 ,

(12)

Let Λ = ∞. We obtain a1⊕···⊕Λ as a vector which satisfies the following equality: L→∞

since separate arguments show that the limit on the right hand side exists.

A.2 The construction of eigenfunctions and their decompositions over analytical defining systems It is convenient to separate the processes into parts. Here, a part without a star stands for the process of eigenfunction construction, and a star-marked part corresponds to the relative part of the decomposition process. For more detailed information on these processes consult [2] and [3]. i Fix i ∈ Ω. If θi and {eip }m p=1 are respectively the measure and the multiplicity sets of 2 i i an ordered representation for Ti then there exists the decomposition L2i = ⊕m p=1 L (ep , θi ),

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p p mi 2 i i which implies Ti = ⊕m p=1 Ti and L (ep , θi ) are Ti -invariant. For vector-operator (⊕i∈Ω ⊕p=1 Tip ) → redesignate → ⊕s Ts , s = {i, p} ∈ Ω1 , we may write Ω1 = ∪Λk=1 Υk . (A) For each Tj , j ∈ Υk and k = 1, Λ, there exists a single cyclic vector aj ∈ L2j and [9, XII.3, Lemma 9 and XIII.5, Theorem 1(I)] a function Wj (xj , λ) defined on Ij × ej (note, that for a fixed i ∈ Ω, Ij = Ii for all p = 1, mi ) and measurable relative to dxj × µaj , such R that Wj (xj , λ) = 0, λ ∈ R \ ej and for any bounded ∆ ⊂ ej : ∆ |Wj (xj , λ)|2 dµaj (λ) ∈ L∞ (Ijn ), n ∈ N. Also Z ¡ j ¢ E (∆)Fj (Tj )aj (xj ) = Wj (xj , λ)Fj (λ) dµaj (λ), (13) ∆

for any Fj ∈ L2 (ej , µaj ). On I k = ∨j∈Υk Ij , we construct the vector-function

W k (∨xj , λ) = {W1 (x1 , λ), . . . , Wn (xn , λ), . . . }, P which is obviously measurable relative to d(∨xj ) × µaj . Separate arguments show that this vector-function is a correctly constructed generalized eigenfunction and thus satisfies the statement of the theorem within each Υk . Note that since for all p = 1, mi there exists the equality (τi − λ)Wip = 0 (see [9, XIII.5, Theorem 1]), it is obvious that (⊕j∈Υk τj − λ)W k = 0, where τj = τi for a fixed i and all p = 1, mi . If P (δ(Ti )) ∩ P (δ(Tj )) has zero spectral measures for all i, j ∈ Ω, then 1 Υk : Ω1 = ∪Λk=1 Υk may be constructed such that Υk contains of indices {i, k}, i ∈ Ω, k = 1, maxi {mi }. (B) Consider the set of indices Ω2 = {j ∈ Ω1 : j = {i, 1}, i ∈ Ω}. Construct 2 Υk : Ω2 = ∪Λk=1 Υk . Apply the reasoning used in (A), substituting Ω2 for Ω1 everywhere it occurs. Hence, for each Υk , we find a vector-function W1k (∨xj , λ) which is the solution of the equation (⊕j∈Υk τj − λ)y = 0. Consider W1k and W1s for s 6= k. For ak there exists the decomposition ak = akk ⊕ akk,s (see A.1). This fact induces the decomposition for W1k : k k k W1k = W1,k ⊕W1,k,s . It is clear that, being the restrictions of W1k , the vector-functions W1,k k and W1,k,s are also the solutions of the equation (⊕j∈Υk τj − λ)y = 0. They, along with akk k such that Ukk : L2 (akk ) → L2 (R, µk ) and akk,s , define unitary transformations Ukk and Uk,s k and Uk,s : L2 (akk,s ) → L2 (R, µk,s ) (see Appendix). This implies that the decomposition k k W k = W1,k ⊕ W1,k,s is correct. k s Define as max{W1,k,s , W1,s,k } the vector-function which corresponds to the vector k s k s max{ak,s , as,k } and respectively min{W1,k,s , W1,s,k } as the vector-function which corres k sponds to that ak,s or as,k , which is not maximal of the two. (C) Without loss of generality, suppose that k = 1 and s = 2. From the reasoning presented in Part (A) of this proof, it follows that © 1 ª 1 2 2 Θ1⊕2 = W1,1 ⊕ W1,2 ⊕ max W1,1,2 , W1,2,1 1

is a correctly constructed vector-function satisfying the statement of the theorem for the case T = [⊕j∈A1 Tj ] ⊕ [⊕q∈A2 Tq ]. Apply the above described process to Θ1⊕2 and W13 to 1 obtain the correctly constructed vector-function: © 1⊕2 ª 3 3 Θ11⊕2⊕3 = Θ1⊕2 ⊕ W ⊕ max Θ , W 1,1⊕2 1,3 1,1⊕2,3 1,3,1⊕2 .

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Continuing this process, we finally obtain: Θ1 (∨xi , λ) = © 1⊕···⊕Λ2 −1 ª Λ2 Λ2 2 2 −1 = Θ1⊕···⊕Λ = Θ1⊕···⊕Λ ⊕ W ⊕ max Θ , W 1 1,1⊕···⊕Λ2 −1 1,Λ2 1,1⊕···⊕Λ2 −1,Λ2 1,Λ2 ,1⊕···⊕Λ2 −1 ,

2 is the function which satisfies (analogously to where in the case of Λ2 = ∞, Θ1⊕···⊕Λ 1 (12)): ° °2 ° °2 Z Z ° ° ° L ° j 1⊕···⊕Λ2 i 1⊕···⊕L °[⊕i∈Ω E∆ ° ° ° , ] Θ dθ(λ) = lim [⊕ E ] Θ dθ (λ) (14) L 1 1 ° ° ° j=1 ∆ °

L→∞

∆

∆

for any bounded Borel set ∆, where ¡ ¢ θL (·) = [⊕Lj=1 E j (·)] a1⊕···⊕L , a1⊕···⊕L

is the measure of the ordered representation of the space ⊕Lj=1 L2j . The limit on the right hand side exists and in fact it appears that Z Z 1⊕···⊕L 2 Θ1 dθL (λ) → Θ1⊕···⊕Λ dθ(λ), 1 ∆

∆

as L → ∞. 3 (D) Define Ω3 = {j ∈ Ω1 : j = {i, 2}, i ∈ Ω}. Construct Υk : Ω3 = ∪Λk=1 Υk . Apply processes (B) and (C) of this proof, substituting everywhere Ω3 instead of Ω2 . 3 We obtain a vector-function Θ1⊕···⊕Λ , which is defined on the set ∪i P (ei2 ). But the set 2 s2 also includes the sets where there are non-empty superpositions of δ(Ti ). Therefore, designating 2 1 3 }, . . . , , W1,2,1 , Θ22 = min{W1,1,2 Θ12 = Θ1⊕···⊕Λ 2

© ª Λ2 2 −1 ΘΛ2 2 +1 = min Θ1⊕···⊕Λ 1,1⊕···⊕Λ2 −1,Λ2 , W1,Λ2 ,1⊕···⊕Λ2 −1 ,

we may again use the process (C) to build the vector-function Θ2 (∨xi , λ) defined on s2 and Θ2 (∨xi , λ) = 0 for λ ∈ R \ s2 . Using processes (B), (C), (D), we finally obtain Θm (∨xi , λ).

(A*) Each kernel Wj (xj , λ) from part (A) may be decomposed: Wj (·, λ) =

nj X

αjs (λ)σjs (·, λ),

s=1

where αjs are supposed to equal zero on R \ ej , see [9, p. 1351]. Supplementing the defining systems with zeros where necessary, we obtain: k

W (∨xj , λ) = ⊕j∈Υk Wj (xj , λ) ⊕j∈Υk =

Nk X q=1

nj X

αjs (λ)σjs (xj , λ) =

s=1

⊕j∈Υk αjq (λ)σjq (xj , λ) =

Nk X q=1

αqk (λ)σqk (∨xj , λ),

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where Nk = max nj , αqk (λ) = j∈Υk

X

j∈Υk

αjq (λ) and σqk (∨xj , λ) = ⊕j∈Υk σjq (xj , λ).

Since ej and ek do not intersect almost everywhere for j, k ∈ Ω2 , j 6= k, the series j∈Υk αjq (λ) converges almost everywhere on ∪j∈Ω2 P (ej ). (B*) and (C*) Now look at part (B). There we obtained the decompositions W1k = k k s s 2 W1,k ⊕ W1,k,s and W1s = W1,s . Let us totally order the set {T j }Λj=1 saying that ⊕ W1,s,k k s k s k k s T ¹ T if max{W1,k,s , W1,s,k } = W1,k,s . At that, T ≃ T if and only if T k ¹ T s 2 and T s ¹ T k . According to this, we build ⊕Λj=1 T j , where T j ¹ T j+1 , j = 1, Λ2 − 1 if Λ2 > 2. The obtained vector-operator is obviously equivalent to the initial vector-operator (comprising unordered operators). Note that P

W1k (∨xi , λ)

=

Nk X

k k α1q (λ)σ1q (∨xj , λ)

q=1

and analogously W1s (∨xi , λ) =

Ns X

s s (∨xj , λ). (λ)σ1p α1p

p=1

(D*) Borrowing the designations from (D) and using processes described in (A*) 3 and (C*), we arrive at the decomposition of Θ1⊕···⊕Λ : 2 Θ21⊕···⊕Λ3 =

N 1⊕···⊕Λ X 3

1⊕···⊕Λ3 1⊕···⊕Λ3 α2s (λ)σ2s (∨xj , λ).

s=1

To obtain Θ2 (∨xi , λ), as in (D), we repeat part (C*) for Λ2 2 3 , . . . , ΘΛ2 2 +1 = W1,Λ . , Θ22 = W1,2,1 Θ12 = Θ1⊕···⊕Λ 2 2 ,1⊕···⊕Λ2 −1

Finally, in the same way we obtain decompositions for all Θk (∨xi , λ), k = 1, m, which will have the form (5).

Acknowledgment The author would like to thank Professor Ashurov R.R., Professor Alimov Sh.A. and Professor Burenkov V.I. for a lot of fruitful discussions. The work has been supported by the ICTP Affiliated Center grant AC - 84 and the State Committee for Science and Technology of the Republic of Uzbekistan.

References [1] M.S. Sokolov: “An abstract approach to some spectral problems of direct sum differential operators”, Electronic J. Diff. Eq., Vol. 75, (2003), pp. 1–6.

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[2] M.S. Sokolov: “On some spectral properties of operators generated by multi-interval quasi-differential systems”, Methods Appl. Anal., Vol. 10(4), (2004), pp. 513–532. [3] R.R. Ashurov and M.S. Sokolov: “On spectral resolutions connected with self-adjoint differential vector-operators in a Hilbert space”, Appl. Anal., Vol. 84(6), (2005), pp. 601–616. [4] W.N. Everitt and A. Zettl: “Quasi-differential operators generated by a countable number of expressions on the real line”, Proc. London Math. Soc., Vol. 64(3), (1992), pp. 524–544. [5] W.N. Everitt and L. Markus: “Multi-interval linear ordinary boundary value problems and complex symplectic algebra”, Mem. Am. Math. Soc., Vol. 715, (2001). [6] R.R. Ashurov and W.N. Everitt: “Linear operators generated by a countable number of quasi-differential expressions”, Appl. Anal., Vol. 81(6), (2002), pp. 1405–1425. [7] M.A. Naimark: Linear differential operators, Ungar, New York, 1968. [8] M. Reed and B. Simon: Methods of modern mathematical physics, Vol. 1: Functional Analysis, Academic Press, New York, 1972. [9] N. Dunford and J.T. Schwartz: Interscience, New York, 1964.

Linear operators, Vol. 2:

Spectral Theory,

CEJM 3(4) 2005 644–653

Centers in domains with quadratic growth Agata Smoktunowicz∗ Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00–956, Warsaw 10, Poland

Received 12 May 2005; accepted 28 July 2005 Abstract: Let F be a field, and let R be a finitely-generated F –algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F –algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F . Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI. c Central European Science Journals. All rights reserved.

Keywords: Centers, domains, growth of algebras, the Gelfand-Kirillov dimension MSC (2000): 16D90, 16P40, 16S80, 16W50

1

Introduction

Bell and Small [4] investigated centralizers in graded domains with quadratic growth. They showed that given a finitely-graded Goldie non-PI domain of Gelfand-Kirillov (GK) dimension 2 over an algebraically-closed field, the centralizer of a non-scalar element of this domain is an affine commutative domain of Gelfand-Kirillov dimension 1. They conjectured that the same holds in the ungraded case. Artin, Schelter and Tate [1] studied central elements in Skylyanian algebras, which are domains of Gelfand-Kirillov dimension 3. Small and Warfield [10] proved that if R is a finitely-generated prime algebra over a field of Gelfand-Kirillov dimension 1 then the center of R is a finitely generated F –algebra of Gelfand-Kirillov dimension 1. Another important result is a theorem of Small, Stafford and Warfield [9] which says that if R is a finitely-generated semiprime algebra of GK dimension 1, then the center of R is a Noetherian domain of GK dimension 1. A result of Zhang [14] says that if R is an affine domain with quotient division algebra Q and A ∗

E-mail: [email protected]

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645

is a commutative subalgebra of Q, then GKdim A ≤ GKdim R where GKdim R denotes the Gelfand-Kirillov dimension of R. For other properties of centralizers in algebras with finite Gelfand-Kirillov dimension we refer to [6, 13]. Given these results it seems natural to look at centers in domains of Gelfand-Kirillov dimension 2. Smith and Zhang [11] showed that if R is a finitely-generated non-PI domain with quotient division ring Q, then the GK dimension of the center of Q is at most GKdim R − 2. Therefore, if R is a finitely-generated non-PI F –algebra which is a domain with quadratic growth and Z is the center of R then GK dimension of Z is 0, hence Z is algebraic over F . The purpose of this paper is to show that either Z is a finitely-generated F –algebra or R is algebraic over F . Recall that given a field F and an affine F –algebra A, the GK dimension of A is defined to be GKdim A = limn→∞ logn (dim (V n )) where V is a finite-dimensional F –subspace of A which contains 1A and generates A as an F –algebra (see [6]). In the case that A is not affine the GK dimension of A is the supremum of the GK dimensions of all affine subalgebras of A. We say that an F –algebra R has quadratic growth if there is a constant c and a generating subspace V of R such that dimF (V + V 2 + · · · + V n ) < cn2 for all n > 0. In particular, GKdim R ≤ 2. Note that R may be not graded. The structure of finitelygraded domains was described in [2]. We state our main result. Theorem 1.1. Let F be a field, and let R be a non-PI affine F –algebra (not necessarily graded) which is a domain with quadratic growth, and let x ∈ R be transcendental over F . Then the centralizer C of x is a P I domain. Moreover, the quotient ring of C is a finitedimensional vector space over F (x), the field of rational functions in the indeterminate x. As a corollary we get Theorem 1.2. Let F be a field, and let R be an affine F –algebra (not necessarily graded) which is a domain with quadratic growth. If the center of R is an infinitely-generated F – algebra, then either R is P I or else R is algebraic over F . It was shown in [12] that Theorem 1.1 and Theorem 1.2 hold provided that F is a finite field.

2

PI algebras

Let F (x) denote the field of rational functions in one indeterminate x over F . We will need the following proposition. Proposition 2.1. (cf. [3]) Let F be a field. Let R be an affine F –algebra which is a domain of GK dimension < 3. If R has a subalgebra A with GKdim A ≥ 2 which satisfies a polynomial identity then R satisfies a polynomial identity.

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Proof. The proof of Proposition 2.1, communicated by Jason Bell [3], can be found in [12, proof of Remark 1].  By F (x, ξ1, ξ2 , . . . , ξi ) we will denote the division F –algebra generated by x, ξ1 , ξ2 , . . . , ξi . Lemma 2.2. Let F be a field, and let R be a finitely-generated non-PI F –algebra, which is a domain with GK dimension smaller than 3. Let x ∈ R be transcendental over F . Let C be the centralizer of x in R, and Q(C) be the quotient ring of C. Then Q(C) = CF (x). Moreover, Q(C) is algebraic over F (x). Proof. A result of Bergman [5] asserts that there are no algebras of GK dimension strictly between 1 and 2. By Proposition 2.1 every element of C is algebraic over F (x). Hence, Q(C) ⊆ CF (x). Since CF (x) ⊆ Q(C), Q(C) = CF (x), and the proof is complete.  Theorem 2.3. Let F be a field, and let R be a finitely-generated non-PI F –algebra, which is a domain with Gelfand-Kirillov dimension smaller than 3. Let x ∈ R be transcendental over F . Then the following are equivalent. (1) There are ξ1 , ξ2 , . . . ∈ C such that ξi+1 ∈ / F (x, ξ1 , ξ2, . . . , ξi ) and ξi ξj = ξj ξi for all i, j > 0. (2) Q(C) is an infinite-dimensional vector space over F (x). Proof. Clearly (1) implies (2). We claim that (2) implies (1). By Lemma 2.2, Q(C) = CF (x). Let K be a maximal subfield of the F (x)–algebra CF (x). It is known [8, Proposition 9.5.2] that if D is a division algebra with center Z, and K is a maximal subfield of D and dimZ K is finite then dimZ D is finite. Since D = CF (x) is a division F (x)–algebra infinite dimensional over the field F (x), it follows that either K is infinite dimensional over Z or Z is infinite dimensional over F (x). Consequently, K is infinite dimensional over F (x). Since K is algebraic over F (x) and K ⊆ CF (x) by Lemma 2.2, it follows that there are ξ1 , ξ2, . . . ∈ C such that ξi+1 ∈ / F (x, ξ1 , ξ2, . . . , ξi ) for every i. This is the desired conclusion. 

3

Assumptions 1–6

In this section, similarly as in [12], we assume that: 1. F is a field, and R is an F –algebra which is a domain with GKdim R < 3. Q(R) is the quotient ring of R (recall that by Jategonkar theorem, a domain of finite Gelfand-Kirillov dimension is automatically an Ore domain [6, p.48]). 2. The algebra R is generated by elements a1 , a2 , . . . , an and x. We denote V = spanF {x, a1 , . . . , an }. 3. We denote R(n) =

Pn

i=0

V i , where V 0 = F .

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4. Element x ∈ R is not algebraic over F . C is the centralizer of x in R, and Q(C) is the quotient ring of C. 5. F (x)V = V F (x). 6. We induce the following order on the generators of R : a1 > a2 > · · · > an > x. Let > be the corresponding deg-lex order on the monoid M generated by a1 , . . . , an and x. The above assumptions will be called Assumptions 1 − 6. By card(S) we denote the cardinality of S. We recall Lemma 9 from [12]. Lemma 3.1. (cf. [12]) Let Assumptions 1 − 6 hold. Then there is a natural number ξ and sets Si ⊆ R(i), consisting of monomials, such that 1. card(Si ) < ξ, for every i ≥ 0. P 2. R(i) ⊆ ij=0 Si F (x) for each i ≥ 0. 3. Every monomial v ∈ R can be written as a finite sum v=

X

ufv,u (x)

u≤v,u∈W,fv,u (x)∈F (x)

where W = S0 ∪ S1 ∪ S2 ∪ . . .. 4. If r ∈ Si and s is a subword of r of degree j then r ∈ Sj . Proof. W is the least subset of M satisfying property 3. For details see [12, proof of Lemma 9]. 

4

Sets W, W1, W2

Definition 4.1. Let w and v be words. We say that v is a beginning of w if w = vu for some word u. Definition 4.2. Let u be a finite word. By deg(u) we denote the length of u. We start with two lemmas derived from Lemma 1.3.13 [7, p.22]. Lemma 4.3. Let w be an infinite word. P (w, i) denotes the number of subwords in w of length i. Suppose that there are α, ν such that P (w, ν) = P (w, ν + 1) = P (w, ν + 2) = · · · = P (w, ν(α + 1)) and P (w, i) ≤ α for every i. Assume that v is a beginning of w and deg(v) ≥ 4(1 + α)v. Then v = pq m for some words p, q of length smaller than 2(1 + α)ν. Moreover, pq i is a beginning of w for every i.

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Proof. The proof is similar to the proof of Lemma 1.3.13 in [7, p. 22]. Divide the beginning of v of length (1 + α)ν onto segments of length ν. Then some two of them are identical. Hence w = u1 u2 u3 u2 u, for some infinite word u and some finite words u1 , u2 , u3 , with deg(u2) = ν, deg(u1u2 u3 u2 ) ≤ (1 + α)ν. By the assumptions, P (w, deg(u2)) = P (w, deg(u2u3 u2)). Since w is an infinite word, it follows that every word of degree deg(u2) is a beginning of exactly one word of length deg(u2u3 u2 ). By the above observation u2 is a prefix of u2 u3 u2 so that w = u1 u2 u3 u2 u3 u2 u3 · · · . Now, we put p = u1 c, q = c′ u2 u3 c for suitable words c, c′ , such that cc′ = u2 u3 .  Lemma 4.4. Let w be an infinite word. P (w, i) denotes the number of subwords in w of length i. Suppose that there is β such that P (w, i) ≤ β for every i. Assume that v is a beginning of w and deg(v) ≥ 4(1 + β)β+5 . Then v = pq m for some words p, q of length smaller than 2(1 + β)β+5 . Moreover, pq i is a beginning of w for every i. Proof. Observe that there is 1 ≤ i ≤ β + 2 such that P (w, (1+β)i) = P (w, (1+β)i +1) = P (w, (1 + β)i + 2) = · · · = P (w, (1 + β)i+1 ). Applying Lemma 4.3, we get the desired conclusion.  Theorem 4.5. Suppose that Assumptions 1 − 6 hold. Let W be as in Lemma 3.1. Then S there is a number τ = τ (W ) such that W = W1 W2 and (1) If v ∈ W1 then v = pq m for some p, q ∈ W , where p, q are monomials of degree smaller than τ and for each natural i, pq i ∈ W1 . (2) If r ∈ W2 then either r = vu where u is a word of length smaller than τ and v ∈ W1 or else r is of length smaller than τ . Proof. Recall that W = S0 ∪ S1 ∪ . . . (see Lemma 3.1). Let W1 = {w ∈ W : for every i ≥ deg(w), w is a beginning of some word from Si }, W2 = {w ∈ W : w ∈ / W1 }. By Lemma 3.1 if w ∈ W then every subword of w is in W . Therefore, for each i there is a natural number n(i) = n(i, W ) such that if u ∈ W2 is of length i, then uv ∈ /W provided that deg(v) ≥ n(i, W ). We claim that if v ∈ W1 then there is an infinite word w such that v is a beginning of w and every finite subword of w belongs to W . Indeed, if v ∈ W1 , then there is c, a word of length 1, such that vc ∈ W1 . Repeating this observation an infinite number times we get the desired infinite word w. By using Lemma 3.1, we see that if u ∈ W1 then every subword of u is in W , hence every finite subword of w is in W . According to Lemma 3.1, w has less than ξ subwords of length i, for each i. Assume that deg(v) ≥ 4(1+ξ)ξ+5. By using Lemma 4.4, observe that v = pq m for some p, q ∈ W , where deg(p), deg(q) < 2(1 + ξ)ξ+5 and for each natural number i, pq i ∈ W1 . Our next step is to investigate the structure of W2 . Denote d = 4(1 + ξ)ξ+5 , and let r ∈ W2 be of length larger than (ξ + 3)(n(d, W ) + d + 1). Let u be the beginning of r of length d. Then u ∈ / W2 . Lemma 3.1 gives u ∈ W , hence m u ∈ W1 . It follows that r = pq for some p, q ∈ W , where p, q are monomials of degree

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smaller than 2(1 + ξ)ξ+5, hence m > ξ + 1. Moreover, for each natural i, pq i ∈ W1 . Let j be the maximal number such that pq j is a beginning of r. Then j > ξ + 1. Consequently, r = pq j c for some word c. Notice that if c has length larger than deg(q)(ξ + 1) then r has more than (ξ + 1) subwords of length deg(q ξ+1) = deg(q)(ξ + 1). On the other hand, Lemma 3.1 gives card(Sdeg(q)(ξ+1) ) < ξ. Therefore deg(c) ≤ deg(q)(ξ + 1) < 2(ξ + 1)ξ+6 . Finally, we set τ = max(2(ξ + 1)ξ+6 , (ξ + 3)(n(d, W ) + d + 1)) where d = 2(1 + ξ)ξ+5 . This finishes the proof. 

5

Combinatorics of words

In this section we suppose that Assumptions 1 −6 hold. Let W, W1 , W2 , τ be as in Lemma 3.1 and Theorem 4.5. We fix elements ξ1 , ξ2 , . . . ∈ C such that ξiξj = ξj ξi for all i, j > 0. Let Fi = F (x, ξ1 , . . . , ξi), the division F –algebra generated by x, ξ1 , . . . , ξi . Definition 5.1. Let t, z be natural numbers, and let v, w ∈ W . We will say that the pair (w, v) is a good pair of type {t, z} if the following holds: (1) w, v ∈ W are words of length smaller than τ and such that wv m ∈ W1 for every m. (2) Let j be the length of wv t . Then wv t ∈ Fz R(j − 1)F (x). Definition 5.2. We say that (w, v) is a good pair if (w, v) is a good pair of type {t, z} for some natural numbers t, z. Lemma 5.3. Let Assumptions 1 − 6 hold, and let ξ be as in Lemma 3.1. If l is a natural number and r ∈ Rl then ξ

xr∈

ξ−1 X i=1

xi rF (x) + F (x)R(l − 1)F (x).

Moreover there is a natural number e, such that for every r ∈ W we have rxe ∈

e−1 X i=1

F (x)rxi + F (x)R(deg(v) − 1)F (x).

Proof. Let r ∈ R. Recall that, from Assumption 5, R(i)F (x) = F (x)R(i), for every i. According to Lemma 3.1, if r1 , . . . rξ ∈ R(ν)F (x) then there are fi (x) ∈ F (x) P not all equal 0 such that ξi=1 ri fi (x) ∈ R(ν − 1)F (x). From this we deduce that if P r ∈ R(ν) then ξi=1 xi rfi (x) ∈ R(ν − 1)F (x) for some fi (x) ∈ F (x) not all equal 0, P i because xi r ∈ R(ν)F (x) for every i, by Assumption 5. Therefore xξ r ∈ ξ−1 i=0 x rF (x) + F (x)R(deg(v) − 1)F (x). Observe that the algebra Rop , with opposite multiplication, also satisfy Assumptions 1 − 6. Applying the above observation for the algebra Rop instead of the algebra R completes the proof.  Lemma 5.4. Let c, b, w, v be words such that (w, v) is a good pair of type {t, z}, for some

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natural numbers t, z. Suppose that c = wv t b. Then c ∈ Fz R(deg(c) − 1)F (x). Proof. Since (w, v) is a good pair of type {t, z}, then wv t ∈ Fz R(j − 1)F (x), where j = deg(wv t). By multiplying by b on the right, we get c = wv t b ∈ Fz R(j − 1)F (x)b ⊆ Fz R(j − 1)F (x)V deg(b) . Assumption 5 yields c ∈ Fz R(j − 1)V deg(b) F (x) ⊆ Fz R(j − 1 + deg(b))F (x). Since deg(c) = deg(wv t) + deg(b) = j + deg(b), the result follows.



Theorem 5.5. Suppose that ξi+1 ∈ / F (x, ξ1 , . . . , ξi ) = Fi for every i > 0. Let w, v be words of length smaller than τ , such that wv i ∈ W for every i (τ is as in Theorem 6.1). Then (w, v) is a good pair. Proof. Let j be a natural number. Fix natural numbers n(j), m(j), p(j) such that: 1. ξj wv m(j) ∈ Fj−1 R(p(j))F (x). 2. n(j) is the length of wv m(j) . 3. Either p(j) < n(j) or p(j) − n(j) is minimal possible. Note that n(j), m(j), p(j) need not be unique. There are three possibilities to consider. (1) n(j) > p(j) for some j. (2) There are j1 , j2 , . . . , jd such that n(ji ) = p(ji ) for every 0 ≤ i ≤ d and d > ξe. Moreover p(ji ) − p(jt ) is divisible by the length of v for all 0 ≤ i, t ≤ d (ξ and e are as in Lemma 3.1 and Lemma 5.3). (3) There are j1 , j2 , . . . , jd such that n(ji ) < p(ji ) for every 1 ≤ i ≤ d and d > ξe. Moreover p(ji ) − p(jt ) is divisible by the length of v for all 1 ≤ i, t ≤ d. Suppose first that Case (1) holds. Let j be such that n(j) > p(j). Since ξj wv m(j) ∈ Fj−1 R(p(j))F (x), we get wv m(j) ∈ Fj R(p(j))F (x) so that (w, v) is a good pair of type {m(j), j}. Assume now that j1 , j2 , . . . , jd are either as in Case (2) or as in Case (3). Denote T = {j1 , j2 , . . . , jd }. Let n ¯ be a natural number such that n ¯ > p(t) and deg(v) divide n ¯ − p(t) m(t) for every t ∈ T . Let t ∈ T . Multiplying the equation ξt wv ∈ Ft−1 R(p(t))F (x) by v c(t) n ¯ −p(t) where c(t) = deg(v) we get ξt wv m(t)+c(t) ∈ Ft−1 R(p(t))F (x)v c(t) . Now v c(t) ∈ V n¯ −p(t) gives ξt wv m(t)+c(t) ∈ Ft−1 R(p(t))F (x)V n¯ −p(t) . Observe that ξt wv m(t)+c(t) ∈ Ft−1 R(p(t))V n¯ −p(t) · F (x) by Assumption 5; and so ξt wv m(t)+c(t) ∈ Ft−1 R(¯ n)F (x). By Lemma 3.1, R(¯ n)F (x) = Pn¯ m(t)+c(t) n − 1)F (x) + Sn¯ F (x) so that ξt wv ∈ Ft−1 R(¯ n − 1)F (x) + i=0 Si F (x) = R(¯ Ft−1 Sn¯ F (x). Notice that there is g(x) ∈ F [x] such that for every t ∈ T , we have 1 ξt wv m(t)+c(t) ∈ Ft−1 R(¯ n − 1)F (x) + Ft−1 Sn¯ F [x] g(x) . Lemma 5.3 yields ξt wv m(t)+c(t) ∈ Pe−1 1 1 Ft−1 R(¯ n − 1)F (x) + i=0 Ft−1 Sn¯ g(x) xi . Denote S = {sxi g(x) : s ∈ Sn¯ , 0 ≤ i ≤ e − 1}. m(t)+c(t) Write ξt wv = bt + dt , where bt ∈ Ft−1 R(¯ n − 1)F (x) and dt ∈ Ft−1 S. Notice that

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card(S) < ξe. Since dt ∈ Ft−1 S for every t ∈ T , and card(T ) = d > ξe > card(S), P it follows that there is ω ∈ T and mi ∈ Fω−1 such that dω = i<ω,i∈T mi di . Now P m(t)+c(t) m(ω)+c(ω) m(i)+c(i) ξt wv = bt +dt yields ξω wv − i<ω,i∈T mi ξi wv ∈ Fω−1 R(¯ n −1)F (x). m(i)+c(i) If Case (2) holds and if i ∈ T then n(i) = p(i); and so deg(wv ) = n(j) + P m(ω)+c(ω) n ¯ − p(j) = n ¯ . Therefore (ξω − i<ω,i∈T mi ξi )wv ∈ Fω−1 R(¯ n − 1)F (x). Notice P that ξω − i<ω,i∈T mi ξi 6= 0, since mi ∈ Fω−1 and ξω ∈ / Fω−1 . Hence wv m(ω)+c(ω) ∈ Fω R(¯ n − 1)F (x). It follows that (w, v) is a good pair of type {m(ω) + c(ω), ω}, since deg(wv m(ω)+c(ω) ) = n ¯. If Case (3) holds and if i ∈ T then n(i) < p(i); and so deg(wv m(i)+c(i) ) = n(i) + n ¯− m(ω)+c(ω) p(i) < n ¯ . Consequently, mi ∈ Fω−1 yields ξω wv ∈ Fω−1 R(¯ n − 1)F (x). Thus, m(ω)+c(ω) (¯ n − 1) − deg(wv ) = p(ω) − n(ω) − 1 < p(ω) − n(ω), contradicting assumption from the beginning of the proof that p(ω) − n(ω) is minimal possible. 

6

The main result

Theorem 6.1. Let Assumptions 1 − 6 hold. Then either Q(C) is a finite-dimensional vector space over F (x) or the algebra R satisfies a polynomial identity. Proof. Assume on the contrary that Q(C) is infinite dimensional over F (x). Then there are ξ1 , ξ2 , . . . ∈ C such that ξiξj = ξj ξi and ξi+1 ∈ / Fi for all i, j > 0 where Fi = F (x, ξ1, . . . , ξi ), the division F –algebra generated by x, ξ1 , . . . , ξi (by Theorem 2.3). Let W be as in Lemma 3.1, and let τ = τ (W ) be as in Theorem 4.5. Notice that there is a finite number of monomials p, q ∈ W such that deg(p), deg(q) < τ . According to Theorem 5.5, there are natural numbers t, z such that if p, q ∈ W are of length smaller than τ and such that pq i ∈ W for every i then (p, q) is a good pair of type {t, z}. Denote τ1 = τ t+1 . By Theorem 4.5, if v ∈ W1 and deg(v) > τ1 then v = pq m for some p, q ∈ W such that deg(p), deg(q) < τ and pq i ∈ W for every i. Notice that deg(v) > τ1 gives m > t. Since (p, q) is a good pair of type {t, z}, Lemma 5.4 yields v ∈ Fz R(deg(v) − 1)F (x). By Theorem 4.5, if v ∈ W2 and deg(v) > τ1 + τ then v = rr ′ where r ∈ W1 and deg(r) > τ1 ; and so v ∈ Fz R(deg(v) − 1)F (x), by Lemma 5.4. Applying this observation several times we get R(i) ⊆ Fz R(τ1 + τ )F (x) for every i. Therefore R ⊆ Fz R(τ1 + τ ), because R(τ + τ1 )F (x) = F (x)R(τ + τ1 ) by Assumption 5. We infer that Fz R is a finite-dimensional left vector space over the field Fz . In particular Fz C is a finite-dimensional left vector space over Fz . Suppose that the algebra R is not P I. According to Lemma 2.2, Fz is algebraic over F (x) so that Fz is finite dimensional over F (x). Since Q(C) = F (x)C = CF (x) by Lemma 2.2, and F (x) ⊆ Fz ⊆ Q(C), it follows that Fz C = Q(C). Therefore, Q(C) is finite dimensional over F (x) contradicting the assumptions.  Theorem 6.2. Let F be a field, and let R be a finitely-generated non-PI F –algebra which is a domain with GK dimension smaller than 3. Let x ∈ R be transcendental over F and let C be the centralizer of x in R. Let Q(C) be the quotient ring of C. Assume that t, m

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are natural numbers, and the algebra R is generated by x and elements y1 , y2 , . . . , yt ∈ R. Suppose that there are αi,j ∈ F for 0 ≤ i, j < m (not all αi,j equal 0) such that for each P k ≤ t, we have i,j<m αi,j xi yk xj = 0. Then Q(C) is a finite-dimensional vector space over F (x). Proof. By Theorem 8 from [12] there are a1 , . . . , an ∈ R (for suitable n) such that the algebra R, and x and a1 , . . . , an satisfy Assumptions 1 − 6. Now the result follows by Theorem 6.1.  Proof (of Theorem 1.1). On the contrary, suppose that Q(C) is not finite dimensional over F (x). Then by Theorem 3 there are there are ξ1 , ξ2 , . . . ∈ C such that ξi+1 ∈ / F (x, ξ1 , ξ2 , . . . , ξi ) and ξiξj = ξj ξi for all i, j > 0. F [y, z] denotes the polynomial ring over F in two commuting indeterminates y, z. Let P r ∈ R, and let f (y, z) = i,j<m αi,j y iz j ∈ F [y, z] where αi,j ∈ F . Then we will use the P following notation f ⌈r⌉ = i,j<m αi,j xi rxj ∈ R. Similarly as in the proof of Theorem 1 in [12] we notice that since the algebra R has quadratic growth there is a finite-dimensional generating subspace V of R and a natural number c such that x ∈ V and dimF (V +V 2 +· · ·+V n ) < cn2 for all n > 0. Consequently there are elements r1 , . . . , rn ∈ R such that for every r ∈ R, there are polynomials P f0 (y, z), f1(y, z), . . . , fn (y, z) ∈ F [y, z] such that f0 (y, z) 6= 0 and f0 ⌈r⌉ = nj=1 fj ⌈rj ⌉ and f0 (y, z) 6= 0. We assume that n is minimal hence n ≤ 2c + 2. Notice that, if n = 0 then, by Theorem 6.2, C is finite dimensional over F (x), a contradiction with our P assumptions. Now, there are fi,j (y, z) ∈ F [y, z] such that fi,0 (⌈ξi ri ⌉) = nj=1 fi,j ⌈rj ⌉ and fi,0 (y, z) 6= 0, for all 1 ≤ i ≤ n. Let T be the algebraic closure of F [y] and let t1 , . . . , tn ∈ T be such that ti+1 ∈ / F (y, t1, . . . , ti ) for i = 1, 2, . . . , n and there is a isomorphism G of fields G : F (y, t1, . . . , ti ) → F (x, ξ1 , ξ2 , . . . , ξi) such that G(y) = x and G(ti ) = ξi for i = 1, . . . , n. Consider matrix B = [bi,j ]1≤i,j≤n , where bi,j = fi,j (y, z) if i 6= j and bi,i = fi,i (y, z) − ti fi,0 (y, z), for all 1 ≤ i, j ≤ n. Since ti+1 ∈ / F (y, t1 , . . . , ti ) and fi,j (y, z) ∈ F [y, z] for every i, j, we see that det(B) 6= 0; hence B is invertible. Therefore DB = Q, for some matrix D with coefficients from M[z] where M = F (y, t1, . . . , tn ) and some diagonal matrix Q with non-zero diagonal entries q1 (y, z), q2 (y, z), . . . , qn (y, z) ∈ F [y, z], because t1 , . . . , tn are algebraic over F (y). P Therefore, by multiplying the equations fi,0 (⌈ξi ri ⌉) = nj=1 fi,j ⌈rj ⌉ on the left by some elements from Fn = F (x, ξ1 , . . . , ξn ) and on the right by some elements from F (x), and taking sums of such equations, we get that qi ⌈ri ⌉ = 0 for all i ≤ n. Consequently, by Theorem 6.2, Q(C) is finite dimensional over F (x) contradicting assumption from the beginning of the proof.  Proof (of Theorem 1.2). Let Z be the center of the algebra R. By a result of Smith and Zhang [11], Z is algebraic over F so that every finitely-generated subalgebra of Z is finite dimensional over F . Assume on the contrary that Z is infinite dimensional over F .

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Then there are ξ1 , ξ2 , . . . ∈ Z such that ξiξj = ξj ξi and ξi+1 ∈ / Zi for all i, j > 0 where Zi = F (ξ1 , . . . , ξi), the division F –algebra generated by ξ1 , . . . , ξi. Observe that Zi ⊆ R for all i, since Z is algebraic over F . Assume now that the algebra R is non-PI and x ∈ R is transcendental over F . Let C = {r ∈ R : rx = xr} be the centralizer of x and Q(C) be the quotient ring of C. By Theorem 1.1, Q(C) is a finite-dimensional vector space P over F (x). Therefore, there is a number t such that ξt ∈ t−1 / Zi it j=0 ξj F (x). Since ξi+1 ∈ follows that x is algebraic over Zt so that x is algebraic over F , contradicting assumption that x is transcendental over F . 

Acknowledgment I am grateful to Jason Bell and to the anonymous referees for many helpful suggestions.

References [1] M. Artin, W. Schelter and J. Tate: “The centers of 3-dimensional Skylyanian algebras“, In: Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math., Vol. 15, Academic Press, San Diego, CA, 1994, pp. 1–10. [2] M. Artin and J.T. Stafford: “Noncommutative graded domains with quadratic growth”, Invent. Math., Vol. 122(2), (1995), pp. 231–276. [3] J.P.Bell: private communication. [4] J.P. Bell and L.W. Small: “Centralizers in domains of Gelfand-Kirillov dimension 2“, Bull. London Math. Soc., Vol. 36(6), pp. 779–785. [5] G.M. Bergman: A note of growth functions of algebras and semigroups, mimeographed notes, University of California, Berkeley, 1978. [6] G. Krause and T. Lenagan: Growth of Algebras and Gelfand-Kirillov Dimension, Revised Edition, Graduate Studies in Mathematics, Vol. 22, American Society, Providence, 2000. [7] M.Lothaire, Algebraic Combinatorics of Words, Cambridge University Press 2002. [8] J.C. McConnel and J.C. Robson: Noncommutative Noetherian Rings, Wiley Interscience, Chichester, 1987. [9] L.W. Small, J.T. Stafford and R.B. Warfield Jr: “Affine algebras of Gelfand-Kirillov dimension one are PI”, Math. Proc. Cambridge Phil. Soc., Vol. 97, (1984), pp. 407– 414. [10] L.W. Small and R.B. Warfield, Jr: “Prime affine algebras of Gelfand-Kirillov dimension one”, J. Algebra, Vol. 91, (1984) pp. 384–389. [11] S.P. Smith and J.J. Zhang: “A remark of Gelfand-Kirillov dimension”, Proc.Amer. Math. Soc., Vol. 126(2), (1998), pp. 349–352. [12] A. Smoktunowicz: “On structure of domains with quadratic growth”, J. Algebra, Vol. 289(2), (2005), pp. 365–379. [13] J.T. Stafford and M. Van den Bergh: “Noncommutative curves and noncommutative surfaces”, Bull. Am. Math. Soc., Vol. 38(2), pp. 171–216. [14] J.J. Zhang: “On lower transcendence degree”, Adv. Math., Vol. 139, (1998), pp. 157– 193.

CEJM 3(4) 2005 654–665

Integral representations of unbounded operators by infinitely smooth kernels Igor M. Novitski˘ı∗ Institute for Applied Mathematics Far-Eastern Branch of the Russian Academy of Sciences, 92, Zaparina Street, 680 000 Khabarovsk, Russia

Received 26 March 2005; accepted 10 August 2005 Abstract: In this paper, we prove that every unbounded linear operator satisfying the Korotkov-Weidmann characterization is unitarily equivalent to an integral operator in L2 (R), with a bounded and infinitely smooth Carleman kernel. The established unitary equivalence is implemented by explicitly definable unitary operators. c Central European Science Journals. All rights reserved.

Keywords: Closed linear operator, integral linear operator, Carleman kernel, characterization theorems for integral operators MSC (2000): 47G10, 45P05

1

Introduction and statement of the main result

The characterization theorems for integral operators in L2 spaces [4, 7] say that these operators are models for most linear operators in Hilbert spaces; and, therefore there is a series of important problems in operator theory, including the famous invariant subspace problem, which can be equivalently reformulated to the class of integral operators (see, e.g., Problem 6 in [8, § 3]). On the other hand, in order to be analytically usable, an integral model of an operator should have a kernel being tractable by means of analytical techniques, and our emphasis here is on the bounded, infinitely smooth Carleman kernels on R2 , uniformly approximable by their restrictions to bounded rectangles, which are in turn perfect for applying classical methods of integral equation theory. The apparatus of Fredholm’s minors has been adapted for such type kernels in [13, 14]. In a sense, the present paper can be thought of as a continuation of [12], where, in ∗

E-mail: [email protected]

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particular, it was proved that if a bounded linear operator is unitarily equivalent to an integral operator, then that operator is unitarily equivalent to an integral operator with a Carleman kernel of any given degree of smoothness. Here, this result extends from the bounded to the unbounded operators in the assumption, and is restricted from the finitely differentiable to the infinitely differentiable kernels in the conclusion. The new result can be expressed by saying that if a closed linear operator is unitarily equivalent to an integral operator with a Carleman kernel, then that operator is unitarily equivalent to an integral operator with an infinitely smooth Carleman kernel. To state the result more precisely we need to record some definitions, and some known results along these lines, for comparison. Throughout this paper, H is a complex, separable, infinite-dimensional Hilbert space with norm k·kH and inner product h·, ·iH , and the symbols C, N, and Z, refer to the complex plane, the set of all positive integers, and the set of all integers, respectively. Let C(H) be the set of all closed, linear, densely defined operators in H, let R(H) be the algebra of all bounded linear operators on H, and let Sp (H) be the Schatten-von Neumann p-ideal of compact linear operators on H [2, Chapter III, §7]. For an operator S in C(H), DS stands for a linear manifold that is the domain of S, and S ∗ for the adjoint to S (w.r.t. h·, ·iH ). We let C0 (H) denote the collection of all those operators S in C(H) for which there exists an orthonormal sequence {en } ⊂ DS ∗ such that lim kS ∗ en kH = 0. n→∞

Let R be the real line (−∞, +∞) equipped with the Lebesgue measure, and let L2 = L2 (R) be the Hilbert space of (equivalence classes of) measurable complex-valued R functions on R equipped with the inner product hf, giL2 = R f (s)g(s) ds and the norm 1

kf kL2 = hf, f iL2 2 . An operator T ∈ C(L2 ) is said to be integral if there exists a measurable function T : R2 → C, a kernel, such that, for each f ∈ DT , Z (T f )(s) = T (s, t)f (t) dt for almost every s in R. R

A kernel T on R2 is said to be Carleman if T (s, ·) ∈ L2 for almost every fixed s in R. Every Carleman kernel, T , induces a Carleman function t from R to L2 by t(s) = T (s, ·) for all s in R for which T (s, ·) ∈ L2 . The existence question for integral operators which are not in C0 (L2 ) seems to be open at present (see Problem 1 in [8, § 5]). The question has the negative answer for integral operators whose domains are L2 (bounded integral operators) and for integral operators whose kernels are Carleman (Carleman integral operators), so both the classes indicated are included in C0 (L2 ). On the other hand, a characterization theorem due to Korotkov [5] and Weidmann [18] asserts that unitary transformations can be constructed to bring operators of C0 (L2 ) to integral form, with the representing kernels being at least simply Carleman. Here is a two-space version of that theorem. Proposition 1.1. If S ∈ C0 (H), then there exists a unitary operator U : H → L2 such that the operator T = USU −1 (DT = UDS ) is an integral operator with a Carleman kernel.

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The main point we are interested here is to try to adjust the unitary operators U in Proposition 1.1 so that the Carleman kernels of the output operators T are SK ∞ -kernels, which we are going to define later. Fix any non-negative integer m and impose on a Carleman kernel T the following smoothness conditions: (i) the function T and all its partial derivatives on R2 up to order m are in C (R2 , C), (ii) the Carleman function t, t(s) = T (s, ·), and all its (strong) derivatives on R up to order m are in C (R, L2 ). Throughout this paper, C(X, B), where B is a Banach space (with norm k·kB ), denotes the Banach space (with the norm kf kC(X,B) = supx∈X kf (x)kB ) of continuous B-valued functions defined on a locally compact space X and vanishing at infinity (i.e., given any f ∈ C(X, B) and ε > 0, there exists a compact subset X(ε, f ) ⊂ X such that kf (x)kB < ε whenever x 6∈ X(ε, f )). A function T that satisfies conditions (i), (ii) is called an SK m -kernel [12]. In addition, an SK m -kernel T is called a K m -kernel [9, 11] if the conjugate transpose function T ∗ (T ∗ (s, t) = T (t, s)) is also an SK m -kernel, i.e., (ii)∗ the Carleman function t∗ , t∗ (s) = T ∗ (s, ·), and all its (strong) derivatives on R up to order m are in C (R, L2 ). Note, incidentally, that Conditions (i), (ii), and (ii)∗ , do not depend on each other in general, and that Condition (i) rules out the possibility for the nonzero SK m -kernels to be depending, for example, on only the difference or the sum of arguments; there are also less trivial examples of inadmissible dependences. The scale of the SK m (K m )-kernels can naturally be supplemented by infinitely smooth Carleman kernels, as follows. Definition 1.2. We say that a function T is an SK ∞ (K ∞ )-kernel if T is an SK m (K m )kernel for each non-negative integer m. What follows is a thorough exposition in the chronological order of what has been published about integral representability of linear operators by K m - and SK m -kernels, whenever m is a priori fixed non-negative integer. Proposition 1.3. Let m be a fixed non-negative integer. Then (A) if for an operator S ∈ C0 (H) there exist a dense in H linear manifold D and an orthonormal sequence {en } such that {en } ⊂ D ⊂ DS ∩ DS ∗ ,

lim kSen kH = 0,

n→∞

lim kS ∗ en kH = 0,

n→∞

−1 then there exists a unitary operator Um : H → L2 such that T = Um SUm (DT = Um DS ) m is an integral operator with a K -kernel (cf. [9], [10]), (B) if for an operator family {Sα : α ∈ A} ⊂ C0 (H) ∩ R(H) there exists an orthonormal sequence {en } such that

lim sup kSα∗ en kH = 0,

n→∞ α∈A

lim sup kSα en kH = 0,

n→∞ α∈A

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then there exists a unitary operator Um : H → L2 such that for each α ∈ A the operator −1 Tα = Um Sα Um (DTα = L2 ) is an integral operator with a K m -kernel of Mercer type (cf. [11]), (C) if for a countable family {Sr : r ∈ N} ⊂ C0 (H)∩R(H) there exists an orthonormal sequence {en } such that sup kSr∗ en kH → 0 as n → ∞, r∈N

then there exists a unitary operator Um : H → L2 such that for each r ∈ N the operator −1 Tr = Um Sr Um (DTr = L2 ) is an integral operator with an SK m -kernel (cf. [12]). It should be mentioned that the converse of (A), as well as of the singleton case of (B), also holds, and is due to a characterization theorem by Korotkov [6] for the so-called bi-Carleman integral operators. One more detail on Proposition 1.3 is that the proof of (C) given in [12] is not direct, and goes by a circuitous route, via Statements (B) and (A). Now we are ready to state our main result, which aims at the desirable sharpening of the conclusion of Proposition 1.1. Theorem 1.4. If S ∈ C0 (H), then there exists a unitary operator U∞ : H → L2 such that −1 the operator T = U∞ SU∞ (DT = U∞ DS ) is an integral operator with an SK ∞ -kernel. The result also refines the singleton case of Statement (C) of Proposition 1.3 by yielding the deeper conclusion under the weaker hypothesis. The next section presents a direct proof of Theorem 1.4, a proof that, unlike that of (C), relies on none of the foregoing smoothing results. It should also be emphasized that the proof below defines in an explicit way that unitary operator whose existence Theorem 1.4 guarantees, by sending a concrete orthonormal basis into another one, no spectral properties of S other than S ∈ C0 (H) are used. Remark 1.5. In our manuscripts [16, 15, 17], we demonstrated that with some adaptations a technique, which we will develop in the next section, works just as well for restricting the conclusions of Statements (A), (B), and (C), to the case m = ∞, under the same assumptions on the input operators, the proofs are also independent of each other and direct. Thus, the integral representations of bounded and unbounded linear operators by means of SK ∞ -kernels and K ∞ -kernels are completely studied. Section 2 concludes with a short note concerning a further development of Theorem 1.4.

2

Proof of Theorem 1.4

In order to make the proof less intricate, we break it up into three “algorithmic” steps. The first step is a geometric preparation for the next two steps. In this step we choose

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in a suitable way an orthonormal basis {fn } for H and split the operator S ∈ C0 (H) to construct four auxiliary operators Q, J, Γ , and A. The second step uses the constructed operators and the basis {fn } to give first a general description and then an explicit example of an orthonormal basis {un } ⊂ L2 of infinitely smooth functions on R. The principal result of the step is a unitary operator from H to L2 , which sends the basis {fn } onto the basis {un } and is suggested as U∞ in the theorem. The rest of the proof (Step 3) is a straightforward verification that the constructed unitary operator does indeed carry S onto an integral operator having an SK ∞ -kernel.

Step 1. Preparing If S ∈ C0 (H), then, by definition, there is an orthonormal sequence {ek } ⊂ DS ∗ such that kS ∗ ek kH → 0 as k → ∞. Choosing a subsequence we may and do assume that X k

(the sum notation

P

1

4 kS ∗ ek kH <∞

(1)

will always be used instead of the more detailed symbol

k

∞ P

).

k=1

Let H be the closed linear span of the ek ’s, and let H ⊥ be the orthogonal complement of H in H. Assume, with no loss of generality, that dim H ⊥ = dim H = ∞. Prove that H ⊂ DS ∗ . Indeed, if f =

P k

hf, ek iH ek ∈ H, then, by (1), the series

(2) P k

hf, ek iH S ∗ ek converges in H;

since S ∗ is closed, the vector f does belong to DS ∗ . If E is the orthogonal projection of H onto H and I is the identity operator on H,  then the subset (I − E)DS ∗ of DS ∗ is a dense subset in H ⊥ . Then choose e⊥ to be an k ⊥ orthonormal basis for the subspace H , with the property  ⊥ ek ⊂ (I − E)DS ∗ , (3) and let {fn } be any orthonormal basis for H such that  {fn } = {ek } ∪ e⊥ k .

(4)

S = (I − E)S + ES.

(5)

Split the operator S as follows

In view of (3) the first summand Q = (I − E)S admits the representation X

 ⊥ X

 ⊥ Qg = Qg, e⊥ g, S ∗ e⊥ on each g ∈ DQ = DS . k H ek = k H ek k

(6)

k

Since E ∈ R(H), the adjoint of the second summand in (5) is given by (ES)∗ = S ∗ E, and is, by (2), in R(H).

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For each f ∈ DS ∗ , let z(f ) = kS ∗ f kH + 1, and define an operator Λ ∈ R(H) by Λ=

X k

Since, for each f ∈ H, the series X k

 ⊥ 1
·, e⊥ k H ek . ⊥ kz ek

(7)

⊥ ∗ ⊥ 1
f, ek H S ek kz e⊥ k

converges in H, it follows that the domain DS ∗ includes the range of Λ, and therefore that the operator S ∗ Λ is in R(H). Now, use the basis {fn } to check that the operator J = S ∗ E is in S 1 (H), and that 4 the operator Γ = S ∗ Λ is in S2 (H): X n

X n

1

4 kJfn kH =

kΓ fn k2H =

X n

X n

1

4 kS ∗ Efn kH =

kS ∗ Λfn k2H =

X k

X

k2

k

1

4 kS ∗ ek kH < ∞,

∗ ⊥ 2

S e π2 k H < .




S ∗ e⊥ + 1 2 6 k H

In particular, it follows that the set {sn } of the s-numbers of J has the property that X n

If J =

P n

1

sn2 < ∞.

(8)

sn h·, pn iH qn is the Schmidt decomposition for J, then the closedness of S makes

it possible to write, for every g ∈ DS , ESg = (ES)∗∗ g = (S ∗ E)∗ g = J ∗ g =

X n

sn hg, qn iH pn ,

(9)

so that (5) becomes S = Q + J ∗.

(10)

Define one more auxiliary operator A ∈ S1 (H) by A=

X n

1

sn4 h·, pn iH qn ,

(11)

and apply (twice) the Schwarz inequality to infer that if kf kH = 1 then kAf kH = ∗

kA f kH =

s X n

s X n

1 2

1

∗ 18 4 sn |hf, pn iH | = (J J) f ≤ kJf kH , 2

H

1 2

1 1

∗ 8 4 sn |hf, qn iH | = (JJ ) f ≤ kJ ∗ f kH . 2

H

(12)

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Step 2. Defining a unitary U∞ In this step, we construct a candidate for the desired unitary operator U∞ in the theorem. Notation 2.1. If an equivalence class f ∈ L2 contains a function belonging to C(R, C), then we shall use [f ] to denote that function. For each h ∈ H, let 1

1

4 4 d(h) = kJhkH + kJ ∗ hkH + kΓ ∗ hkH ,

(13)

and note that the compactness of each of J and Γ , proved in the previous step, implies that d(ek ) → 0 as k → ∞. (14) Take any orthonormal basis {un }∞ n=1 for L2 , with the properties: (a) for each i and for each n ∈ N, the ith derivative, [un ](i) , of [un ] is in C(R, C) (here and throughout, the letter i is reserved for all non-negative integers), ∞ ∞ (b) the set {un } is the disjoint union

of two sequences {gk }k=1 and {hk }k=1 such that if



, then, for each i, Hk,i = [hk ](i) , Gk,i = [gk ](i) C(R,C)

C(R,C)

X

Hk,i < ∞,

(15)


kz e⊥ k Hn(k),i < ∞,

(16)

k

X k

X k

d(xk ) (Gk,i + 1) < ∞,

(17)

∞ where {n(k)}∞ k=1 is a strictly increasing sequence of positive integers, and {xk }k=1 is a subsequence of the sequence {ek }.

Example 2.2. A good example of such a basis {un } can be adopted from the wavelet theory, as follows. Let ψ be the Lemari´e-Meyer wavelet, Z 1 1 [ψ] (s) = eıξ( 2 +s) sgn ξb(|ξ|) dξ (s ∈ R) (18) 2π R with the bell function b being infinitely smooth and compactly supported on [0, +∞) (see, e.g., [1, § 4] or [3, Example D, p. 62] for details). Then [ψ] is of the Schwartz class S(R), so its every derivative [ψ](i) is in C(R, C). In addition, the “mother wavelet” ψ generates an orthonormal basis {ψjk }j, k∈Z for L2 by j

ψjk = 2 2 ψ(2j · −k) (j, k ∈ Z). In a completely arbitrary manner, rearrange the two-indexed set {ψjk }j, k∈Z into a simple sequence so that it becomes {un }∞ n=1 ; and, reveal that the latter has the property

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(b). Indeed, if, in view of that rearrangement, un = ψjn kn whenever n ∈ N, then, for each i,



(i) (i) = [ψjn kn ] ≤ Dn Ai for all n ∈ N,

[un ] C(R,C)

where

C(R,C)

 2   2j n Dn =  1 |jn |   √ 2

if jn > 0,

if jn ≤ 0,

2 (i) i+ 21 ) ( Ai = 2

[ψ]

C(R,C)

.

If a strictly increasing sequence {l(k)}∞ of positive integers satisfies jl(k) → −∞ as k=1  ∞  ∞ k → ∞, then split {un } into hk = ul(k) k=1 and gk = um(k) k=1, with {m(k)}∞ k=1 = ∞ N \ {l(k)}k=1 , and observe that X Dl(k) < ∞. (19) k

Then, for each i, the sums in (15), (16), and (17), are bounded by X X X
 Ai Dl(k) , Ai kz e⊥ D , and (A + 1) d(x ) max 1, D , l(n(k)) i k m(k) k k

k

respectively, where the last-written two expressions can always be made finite by an appropriate choice of subsequences {n(k)}∞ k=1 of N and {xk } of {ek } (see (19), (14)).   Let us return to the proof of the theorem. Let x⊥ = e⊥ ∪ ({ek } \ {xk }), and k k  ⊥ observe via (4) that {fn } = {xk } ∪ xk . Define a unitary operator U∞ : H → L2 on the basis vectors by setting U∞ x⊥ k = hk ,

U∞ xk = gk ,

for all k ∈ N,

(20)

in the harmless assumption that, for each k ∈ N, U∞ fk = uk ,

U∞ e⊥ k = hn(k) ,

(21)

where {n(k)} is just that sequence which occurs in (16).

Step 3. Verifying −1 In this step, we are to verify that T = U∞ SU∞ (DT = U∞ DS ) is an integral operator ∞ having an SK -kernel. For this purpose, in view of the splitting (10), it suffices to −1 −1 check that the operators P = U∞ QU∞ (DP = DT ) and F = U∞ J ∗ U∞ (DF = L2 ) are ∞ integral operators having SK -kernels. The verification of the latter properties goes by representing all pertinent functions as infinitely smooth sums of termwise differentiable series of infinitely smooth functions. Combine (6) with (21) to infer that X

 Pf = f, T ∗ hn(k) L2 hn(k) for all f ∈ DP , (22) k

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where, by (7), T ∗ hn(k) =

X


X ∗ ⊥  ⊥ S ∗ e⊥ , f u = kz e S Λek , fn H un n H n k k n

n

= kz e⊥ k


X ⊥ ∗  ek , Γ fn H un n

(k ∈ N), (23)

with the series convergent in L2 . Prove that, for any fixed i, the series X

 (i) ∗ e⊥ (s) (k ∈ N) k , Γ fn H [un ] n

converge in the space C(R, C). Indeed, all these series are pointwise dominated on R by one series X kΓ ∗ fn kH [un ](i) (s) , n

which converges uniformly in R because its component subseries X X

(i)

Γ ∗ x⊥

[h ] (s) kΓ ∗ xk kH [gk ](i) (s) , k H k k

k

are in turn dominated by the convergent series X X d(xk )Gk,i, kΓ ∗ k Hk,i, k

k

respectively (see (20), (13), (17), (15)). For (23), the reasoning just given implies in turn that, for each k ∈ N,

 (i)


∗

≤ Ci kz e⊥ (24)

T hn(k) k , C(R,C)

with a constant Ci independent of k. From (23), it also follows that

∗

T hn(k)

L2


≤ kz e⊥ k kΓ k

(k ∈ N).

If a function P : R2 → C and a Carleman function p : R → L2 are defined as X    P (s, t) = hn(k) (s) T ∗ hn(k) (t), k

p(s) = P (s, ·) =

X k

 hn(k) (s)T ∗ hn(k)

whenever s, t ∈ R, then, for all non-negative integers i and j, X (i)  (j) ∂ i+j P (s, t) = hn(k) (s) T ∗ hn(k) (t), i j ∂s ∂t k X (i) di p (s) = hn(k) (s)T ∗ hn(k) , i ds k

(25)

(26)

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because, in view of (24), (25), and (16), the series just displayed converge (and even absolutely) in C (R2 , C) and C (R, L2 ), respectively. Thus,
∂ i+j P 2 ∈ C R , C , ∂si ∂tj

di p ∈ C (R, L2 ) dsi

(27)

whenever i and j are non-negative integers. From (25) and (16), it follows that the series (22) (viewed, of course, as a series with terms belonging to C(R, C)) converges and even absolutely in the C(R, C) norm, and therefore that its pointwise sum is nothing else than [P f ]. On the other hand, the established properties of the series in (26) make it possible to write, for each temporarily fixed s ∈ R, the following chain of relations * + X

X     f, T ∗ hn(k) L2 hn(k) (s) = f, hn(k) (s)T ∗ hn(k) k

k

=

Z

R

X k



hn(k) (s)



T ∗ hn(k)



!L2

(t) f (t) dt =

Z

P (s, t)f (t) dt R

whenever f is in DP . This together with (27) imply that P : DP → L2 is an integral operator with the SK ∞ -kernel P . The next thing is to observe that the inducing kernel of the integral operator F = −1 U∞ J ∗ U∞ ∈ S 1 (L2 ) is the sum of the bilinear series 4

X

1 2

sn U∞ A∗ qn (s)U∞ Apn (t)

=

n

X

sn U∞ pn (s)U∞ qn (t)

n

!

(28)

in the sense of almost everywhere convergence on R2 (see (9), (11)). The functions used in (28) can be written as the series X X U∞ Apk = hpk , A∗ fn iH un , U∞ A∗ qk = hqk , Afn iH un (k ∈ N), n

n

convergent in L2 . Show that, for any fixed i, the functions [U∞ Apk ](i) , [U∞ A∗ qk ](i) (k ∈ N) make sense, are all in C(R, C), and their C(R, C) norms are bounded independent of k. Indeed, all the series X X hpk , A∗ fn iH [un ](i) (s), hqk , Afn iH [un ](i) (s) (k ∈ N) n

n

are dominated by one series X n

(kA∗ fn kH + kAfn kH ) [un ](i) (s)

that converges uniformly on R, because it consists of the two uniformly convergent subseries on R X X

⊥
(i)

A∗ x⊥

+ Ax [h ] (s) (kA∗ xk kH + kAxk kH ) [gk ](i) (s) , , k k H k H k

k

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dominated by the convergent series X d(xk )Gk,i, k

X k

2 kAk Hk,i,

respectively (see (13), (12), (17), (15)). Now define a function F : R2 → C and a Carleman function f : R → L2 by X 1 F (s, t) = sn2 [U∞ A∗ qn ] (s)[U∞ Apn ] (t), n

f (s) = F (s, ·) =

X

1

(29)

sn2 [U∞ A∗ qn ] (s)U∞ Apn ,

n

whenever s, t ∈ R (cf. (28)). Then, for all non-negative integers i, j and all s, t ∈ R, X 1 ∂ i+j F (s, t) = sn2 [U∞ A∗ qn ](i) (s)[U∞ Apn ](j) (t), ∂si ∂tj n X 1 di f (s) = sn2 [U∞ A∗ qn ](i) (s)U∞ Apn , dsi n as the series just written converge in C (R2 , C) and C (R, L2 ), respectively, due to (8). Therefore, it follows that F is the SK ∞ -kernel of F . In accordance with (10), we have T = P + F , and hence, for each f ∈ DT , Z Z Z (T f )(s) = P (s, t)f (t) dt + F (s, t)f (t) dt = (P (s, t) + F (s, t))f (t) dt R

R

R

for almost every s in R. Therefore T : DT → L2 is an integral operator, and that kernel T of T , which is defined by T (s, t) = P (s, t) + F (s, t) whenever s and t are in R, inherits the SK ∞ -kernel properties from its terms. Consequently, T has the SK ∞ -kernel T . The proof of the theorem is complete. Remark 2.3. For further research, there is at least one challenging question: Let S ∈ e∞ : H → L2 such that C0 (H). Does it then follow that there exists a unitary operator U e∞ S U e −1 is an integral operator having an SK ∞ -kernel which is real-analytic together U ∞ with its Carleman function? In our opinion, the question is not answered by the unitary operator (20) in general because the series (26), (29) may fail to represent real-analytic functions, in spite of the fact that the function (18) is the restriction of an entire function e∞ , on R. A much more analytic machinery should be used to construct such a unitary U if the latter does indeed exist.

Acknowledgment The author wishes to express his gratitude to the Mathematical Science Division of the Russian Academy of Sciences for its support of this research (grant N 04-1-OMH-079), and to the referees for their remarks.

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References [1] P. Auscher, G. Weiss and M.V. Wickerhauser: “Local Sine and Cosine Bases of Coifman and Meyer and the Construction of Smooth Wavelets”, In: C.K. Chui (Ed.): Wavelets: a tutorial in theory and applications, Academic Press, Boston, 1992, pp. 237–256. [2] I.Ts. Gohberg and M.G. Kre˘ın: Introduction to the theory of linear non-selfadjoint operators in Hilbert space, Nauka, Moscow, 1965. [3] E. Hern´andez and G. Weiss: A first course on wavelets, CRC Press, New York, 1996. [4] P. Halmos and V. Sunder: Bounded integral operators on L2 spaces, Springer, Berlin, 1978. [5] V.B. Korotkov: “Classification and characteristic properties of Carleman operators”, Dokl. Akad. Nauk SSSR, Vol. 190(6), (1970), pp. 1274–1277; English transl.: Soviet Math. Dokl., Vol. 11(1), (1970), pp. 276–279. [6] V.B. Korotkov: “Unitary equivalence of linear operators to bi-Carleman integral operators”, Mat. Zametki, Vol. 30(2), (1981), pp. 255–260; English transl.: Math. Notes, Vol. 30(1-2), (1981), pp. 615–617. [7] V.B. Korotkov: Integral operators, Nauka, Novosibirsk, 1983. [8] V.B. Korotkov: “Some unsolved problems of the theory of integral operators”, In: Sobolev spaces and related problems of analysis, Trudy Inst. Mat., Vol. 31, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 1996, pp. 187–196; English transl.: Siberian Adv. Math., Vol. 7(2), (1997), pp. 5–17. [9] I.M. Novitski˘ı: “Reduction of linear operators in L2 to integral form with smooth kernels”, Dokl. Akad. Nauk SSSR, Vol. 318(5), (1991), pp. 1088– 1091; English transl.: Soviet Math. Dokl., Vol. 43(3), (1991), pp. 874–877. [10] I.M. Novitski˘ı: “Unitary equivalence between linear operators and integral operators with smooth kernels”, Differentsial’nye Uravneniya, Vol. 28(9), (1992), pp. 1608– 1616; English transl.: Differential Equations, Vol. 28(9), (1992), pp. 1329–1337. [11] I.M. Novitski˘ı: “Integral representations of linear operators by smooth Carleman kernels of Mercer type”, Proc. Lond. Math. Soc. (3), Vol. 68(1), (1994), pp. 161–177. [12] I.M. Novitski˘ı: “A note on integral representations of linear operators”, Integral Equations Operator Theory, Vol. 35(1), (1999), pp. 93–104. [13] I.M. Novitski˘ı: “Fredholm minors for completely continuous operators”, Dal’nevost. Mat. Sb., Vol. 7, (1999), pp. 103–122. [14] I.M. Novitski˘ı: “Fredholm formulae for kernels which are linear with respect to parameter”, Dal’nevost. Mat. Zh., Vol. 3(2), (2002), pp. 173–194. [15] I.M. Novitski˘ı: Simultaneous unitary equivalence to bi-Carleman operators with arbitrarily smooth kernels of Mercer type, arXiv:math.SP/0404228, April 12, 2004. [16] I.M. Novitski˘ı: Integral representations of closed operators as bi-Carleman operators with arbitrarily smooth kernels, arXiv:math.SP/0404244, April 13, 2004. [17] I.M. Novitski˘ı: Simultaneous unitary equivalence to Carleman operators with arbitrarily smooth kernels, arXiv:math.SP/0404274, April 15, 2004. [18] J. Weidmann: “Carlemanoperatoren”, Manuscripta Math., Vol. 2 (1970), pp. 1–38.

CEJM 3(4) 2005 666–704

The rate of convergence for spectra of GUE and LUE matrix ensembles∗ Friedrich G¨otze1† , Alexander Tikhomirov2,3 1

Faculty of Mathematics, University of Bielefeld, Germany 2 Faculty of Mathematics, Syktyvkar State University, Syktyvkar, Russia 3 Department of Mathematics, IMM of the Ural Branch RAS, Syktyvkar, Russia

Received 23 December 2004; accepted 19 August 2005 Abstract: We obtain optimal bounds of order O(n−1 ) for the rate of convergence to the semicircle law and to the Marchenko–Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively. c Central European Science Journals. All rights reserved.

Keywords: Random matrix theory, Gaussian unitary ensemble, Laguerre unitary ensemble, Stein’s method, semicircle law, Marchenko–Pastur law MSC (2000): 60F05, 33E05, 33E15

1

Introduction

We shall study spectral asymptotics for two classes of random matrices. The first class is the Gaussian unitary ensemble (GUE), and the second class is the complex Wishart unitary ensemble, which is also called Laguerre unitary ensemble (LUE). ∗

Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, by RFBR grant N 02-01-00233, and by RFBR–DFG grant N 04-01-04000. † E-mail: [email protected]

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1.1 The GUE We start with the GUE. Let Xjk and Yjk , 1 ≤ j ≤ k < ∞, be two independent triangular 2 arrays of independent Gaussian random variables with EXjk = EYjk = 0 and EXjk = 1 1 2 (1 + δjk ), EYjk = 2 (1 − δjk ), where δjk denotes Kronecker’s symbol. Let Xkj = Xjk , 2 Ykj = −Yjk for 1 ≤ j < k < ∞. For a fixed n ≥ 1, we denote by λ1 ≤ . . . ≤ λn the eigenvalues of the unitary n × n matrix Wn = (Wn (j, k))nj,k=1,

1 Wn (j, k) = √ (Xjk + iYjk ), for 1 ≤ j ≤ k ≤ n, n

(1.1)

and define its empirical spectral distribution function by n

1X Fn (x) = I{λ ≤x} , n j=1 j

(1.2)

where I{B} is the indicator of an event B. We investigate the rate of convergence of the expected spectral distribution function EFn (x) to the distribution function of Wigner’s semicircle law. Let g(x) and G(x) denote the density and the distribution function of the standard semicircle law respectively, that is Z x 1√ g(x) = 4 − x2 I{|x|≤2}, G(x) = g(u)du. (1.3) 2π −∞ The aim of this paper is to prove an optimal bound for the Kolmogorov distance between EFn (x) and G(x), that is ∆n = sup |EFn (x) − G(x)|,

(1.4)

x

for the matrices from the GUE. The main result is the following Theorem 1.1. There exists an absolute positive constant C such that, for any n ≥ 1, ∆n ≤ Cn−1 .

(1.5)

The convergence of the spectral distribution of matrices from the GUE has been studied in the pioneering work by Wigner (cf. [26, 11, 7]). The rate of convergence of these spectral distributions for the GUE has been studied by G¨otze and Tikhomirov [16] 2 who proved that ∆n = O(n− 3 ). The rate of convergence for the expected spectral distribution functions of real symmetric or Hermitean random matrices with independent and not necessarily Gaussian entries – the so-called Wigner ensemble – has been studied by several authors. In 1993, Bai [2] conjectured that the optimal bound for ∆n for Wigner matrices should be of order O(n−1). Bai et al. [2, 4] proved that ∆n = O(n−1/4 ) and that ∆n = O(n−1/3 ). The bound O(n−1/2 ) assuming uniform bounded fourth moments has been shown by

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G¨otze and Tikhomirov in [17]. Bai et al. [5] proved that ∆n = O(n−1/2 ) assuming that supi,j,n E|Xij |8 < ∞. Girko [12] proved a bound for the difference between the Stieltjes transforms of the distribution functions EFn (x) and G(x) which yields bounds for ∆n of order O(n−1/6 ) (see inequality (3.12) in [12] and e.g. inequality (4.26) in [2]). Girko [13] stated as well that ∆n = O(n−1/2 ) assuming uniform bounded fourth moments. He used a different approach to that in Bai [2] or the one in our paper [17]. Girko [14] states again ∆n = O(n−1/2 ) providing an update of his previous proof with extended arguments and some corrections. We consider as well the rate of convergence of the density of the expected spectral distribution function of random matrices from GUE. In [17], it has been shown that there 1 1 exist some positive constants γ and C such that, for x ∈ [−2 + γn− 3 , 2 − γn− 3 ], |pn (x) − g(x)| ≤

C . n(4 − x2 )2

(1.6)

In this paper we shall prove a more precise result. Theorem 1.2. There exist absolute positive constants, γ and C, such that, for x ∈ 2 2 [−2 + γn− 3 , 2 − γn− 3 ], C . (1.7) |pn (x) − g(x)| ≤ n(4 − x2 ) The GUE plays a significant role in the random matrix theory. We consider this ensemble as the special Wigner ensemble (matrices with independent entries). The GUE is a special case of a general unitary invariant matrix ensemble with density of type exp{−Tr V (Wn )}, where V (x) is an even order polynomial of x. Such ensembles have been intensively studied. In the case of GUE the following representation for the expected spectral density holds r  r  n−1 1 X 2 n pn (x) = ϕk x (1.8) 2n k=0 2

where ϕk (x) denote the Hermitean orthogonal functions. (See, for example, Mehta [26], p. 420). Analogous formulas with coefficients depending on the weight function V (x) and with orthogonal functions ϕk (x) with respect to exp{− 21 V (x)} hold for matrices from general unitary invariant ensembles. In the last years the asymptotic behavior of the orthogonal functions ϕk (x) has been studied for instance via saddle-point methods analyzing solutions of matrix valued Riemann–Hilbert problems. This method has been successfully applied in the random matrix theory by Deift [7], Deift et al. [8]. In particular, in Deift et al. [8] asymptotic expansions for general orthogonal functions ϕk (x) have been obtained. Based on this paper, Ercolani and McLaughlin [9] have shown asymptotic expansions of the spectral density pn (x) for the general weights exp{−nV (x)}. This approach has been successfully used by Gustavsson for GUE in [21] to estimate the expectation of the number of eigenvalues in an interval. This approach may be used to prove Theorem 1.2, using local asymptotic expansions of Hermitean orthogonal functions (which is a particular case of results of Deift et al. [8]) with at least two expansion terms.

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We used an alternative approach for the proof of Theorem 1.1 and 1.2, since the local approach mentioned above involves combining local expansions in the bulk and on the boundary in order to obtain optimal rates for the distribution function. Proving Theorem 1.1 this way would require rather uniform error bounds in Theorem 1.2 when approaching the boundary of the spectrum. Instead we apply results by Haagerup and Thorbjørnson, and arrive at a representation of the density pn (x) as a solution of some linear differential equation of third order. This allows us to prove the results of both Theorem 1.1 and Theorem 1.2, using classical bounds by Erdelyi [10] or Muckenhaupt [27] for the decay of the Hermitean orthogonal functions near the end points of the spectrum. In Remark 2.3 we provide a characterizing equation for pn (x) in the sense of Ch. Stein. For growing n this equation approximates the Stein equation in G¨otze and Tikhomirov [19].

1.2 The LUE Let Xjk and Yjk , j ≥ 1, k ≥ 1, be two independent arrays of independent Gaussian 2 random variables with EXjk = EYjk = 0 and EXjk = 12 (1 + δjk ), EYjk2 = 12 (1 − δjk ). For integers 1 ≤ n ≤ m consider an n × m matrix B = (Bjk ) with entries Bjk = Xjk + iYjk , 1 ≤ j ≤ n and 1 ≤ k ≤ m. Let 1 W = BBT . (1.9) m We say that W belongs to the so-called Laguerre unitary ensemble (LUE) of random matrices. The joint distribution of entries of the matrix W is known as the complex Wishart distribution (see, for instance [15, 23], and references in the last paper). Let λ1 , . . . , λn denote the eigenvalues of the matrix W. Define the spectral distribution function of the matrix W by n

Fn(m) (x)

1X = I{λ ≤x} . n j=1 j

(1.10)

n Assuming m → y ∈ (0, 1] as n → ∞, it is well-known that EFn (x) convergences to the Marchenko–Pastur distribution function My (x) with the density

my (x) =

1 p (x − a)(b − x)I{[a,b]} (x), 2πyx

(1.11)

√ √ where a = (1 − y)2 , b = (1 + y)2 (see [25]). The matrix W is a covariance type matrix which has a Wishart distribution. We shall investigate bounds for ∆(m) = sup |EFn(m) (x) − My (x)|. n

(1.12)

x

The main results of this part of the paper is the following n Theorem 1.3. Let y = m . Assume that m = m(n) is a sequence of integers such that there exist some positive constants a1 and a2 such that 0 < a1 ≤ y ≤ a2 < 1 for all n ≥ 1.

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Then there exists a positive constant C = C(a1 , a2 ) depending on a1 , a2 and such that, for any n ≥ 1, ∆(m) ≤ Cn−1 . (1.13) n (m)

Bounds for ∆n for (non-Gaussian) matrices in the case 0 < a1 ≤ y ≤ a2 < 1 1 (m) have been studied by Bai ([3]), proving ∆n = O(n− 4 ) . Bai at al. [6] and G¨otze and 1 (m) Tikhomirov [18] showed that ∆n = O(n− 2 ). The case y = 1 is more complicated for 5 (n) non-Gaussian matrices. For example, Bai obtained bounds in [3] ∆n = O(n− 48 ) and in 1 (n) [6] ∆n = O(n− 8 ) only. We shall prove the following result for Gaussian matrices: Theorem 1.4. Let m = n. Then there exists an absolute positive constant C such that, for any n ≥ 1, −1 ∆(n) (1.14) n ≤ Cn . For the densities of spectral distribution functions we prove the following results. Theorem 1.5. Let m = n. Then there exist absolute positive constants, γ and C, such 2 that, for n ≥ 1 and for x ∈ [γn−2 , 4 − γn− 3 ], |p(n) n (x) − m1 (x)| ≤

C . nx(4 − x)

(1.15)

For the case y < 1, we shall prove n Theorem 1.6. Let y = m . Assume that m = m(n) is a sequence of integers such that there exist some positive constants a1 , a2 , 0 < a1 ≤ y ≤ a2 < 1, for all n ≥ 1. Then there exist positive constants C = C(a1 , a2 ) and γ = γ(a1 , a2 ) depending on a1 , a2 and 2 2 such that, for n ≥ 1 and x ∈ [a + γn− 3 , b − γn− 3 ],

|pn(m) (x) − my (x)| ≤

C . n(x − a)(b − x)

(m)

(1.16)

In Remark 3.3 we provide a characterizing equation for pn (x) in the sense of Ch. Stein. The proofs of Theorems 1.1–1.4 are based on the representation of the density of expected spectral distribution function as a solution of some linear differential equation. See Lemmas 2.1 and 3.7 below. We also give a Stein type characterization of the expected spectral distribution functions of random matrices from the both GUE and LUE with a third order differential operator respectively. See Remarks 2.3 and 3.3. These equations were obtained from explicit formulas for Laplace transforms of the expected distribution functions for the GUE and LUE in terms of confluent hypergeometric functions, which have been deduced by Haagerup and Thorbjørnsen in [22]. The approach of Haagerup and Thorbjørnson has been used by Ledoux in [24] in order to prove sharp bounds for the largest eigenvalues of matrices from the GUE.

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671

Wigner matrices. Gaussian unitary ensemble (GUE)

In this Section we prove Theorems 1.1 and 1.2. Let pn (x) be the density of the expected spectral distribution function of the matrix W from GUE and let fn (t) be its characteristic function. Haagerup and Thorbjørnsen [22] have shown that  fn (t) = exp −t2 /2n Φ(1 − n, 2; t2 /n), (2.1) where Φ(α, β; s) is the confluent hypergeometric function which defined by the following equality Φ(α, β; s) = 1 +

αs α(α + 1) s2 α(α + 1)(α + 2) s3 + + +··· . β 1! β(β + 1) 2! β(β + 1(β + 2)) 3!

The function Φ(α, β; s) satisfies the following differential equation s

d2 Φ dΦ + (β − s) − αΦ = 0. 2 ds ds

(2.2)

Using (2.1) and (2.2), we derive a differential equation for the density pn (x). Lemma 2.1. The density pn (x) of the expected spectral distribution of a random matrix from GUE satisfies the following differential equation (4 − x2 )p′n (x) + xpn (x) +

1 ′′′ p (x) = 0. n2 n

(2.3)

Remark 2.2. As n tends to infinity this equation may be approximated by (4 − x2 )p′ (x) + xp(x) = 0,

(2.4)

which has been used by G¨otze and Tikhomirov in [19]. Proof. We consider the function

From (2.1) it follows that

 ψ(t) = exp t2 /2n fn (t).

(2.5)

ψ(t) = Φ(1 − n, 2; t2 /n)

(2.6)

and ψ ′ (t) =

2t ′ Φ (1 − n, 2; t2 /n), n

ψ ′′ (t) =

2 ′ 4t2 Φ (1 − n, 2; t2 /n) + 2 Φ′′ (1 − n, 2; t2 /n). (2.7) n n

We rewrite these equalities in the form n ′ ψ (t), 2t n2 1 Φ′′ (1 − n, 2; t2 /n) = 2 (ψ ′′ (t) − ψ ′ (t)). 4t t Φ′ (1 − n, 2; t2 /n) =

(2.8)

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The relation (2.2), (2.6) and (2.8) together imply that tψ ′′ (t) + (3 −

4(1 − n) 2t2 ′ )ψ (t) − tψ(t) = 0. n n

(2.9)

The definition (2.5) yields t exp{t2 /2n}fn (t) + exp{t2 /2n}fn′ (t), n   t2 2t 1 ′′ ψ (t) = + 2 exp{t2 /2n}fn (t) + exp{t2 /2n}fn′ (t) + exp{t2 /2n}fn′′ (t). n n n ψ ′ (t) =

(2.10)

By substitution of (2.10) in (2.9), we get tfn′′ (t)

+

3fn′ (t)

  t3 + 4t − 2 fn (t) = 0. n

(2.11)

Since Φ(1 − n, 2; t2 ) is a polynomial of order n − 1 and fn (t) satisfies the equality (2.1) we may apply an inverse Fourier transform. We get (4 − x2 )p′n (x) + xpn (x) +

1 ′′′ p (x) = 0, n2 n

which proves the lemma.

(2.12) 

2.1 Applications of Lemma 2.1 Let n ≥ 1. We introduce the conjugate differential operator Dn (x) :=

d3 2 2 d + n (4 − x ) − 3n2 x, dx3 dx

(2.13)

and introduce a class of functions f : R 7→ R defined by n Rn := f : R 7→ R : f (3) exists,

o lim |xk f (ν) (x) exp{−nx2 }| = 0, ν = 0, 1, 2, k ≥ 1 . (2.14)

|x|→∞

Remark 2.3. (Stein type characterizing equation for pn (x)) Let n ≥ 1. A random variable ξ has the density pn (x) if and only if, for any function f ∈ Rn EDn (ξ)f (ξ) = 0.

(2.15)

Remark 2.4. (Integrodifferential equation for pn (x)) Using Lemma 2.1, we may write for |x| < 2 √ √ Z 4 − x2 4 − x2 x p′′′ n (u)du pn (0) + pn (x) = (2.16) 3 . 2 2 2 n 0 (4 − u ) 2

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Note that the distribution functions EFn (x) and G(x) are symmetric. This implies EFn (0) = G(0) and Z x EFn (x) − G(x) = (pn (u) − g(u))du. (2.17) 0

The equations (2.16) and (2.17) together imply that, for |x| < 2,  Z x √ 1 1 EFn (x) − G(x) = pn (0) − 4 − u2 du 2 π 0 ! Z u ′′′ Z x√ 1 p (s)ds n + 2 4 − u2 du. 3 n 0 0 (4 − s2 ) 2

(2.18)

We shall also use the representation (1.8). The weight function for the GUE is V (x) = x and ϕk (x), k is positive integer, denotes the Hermitean orthogonal function 2

√
− 1 x2 ϕk (x) = 2k k! π 2 exp{− }Hk (x) 2 k 2 1
√ x d −2 k = 2 k! π exp{ } − exp{−x2 }, 2 dx

(2.19)

where Hk (x) stands for the kth Hermitean polynomial. Using the Christoffel–Darboux formula and the relations between Hermitean orthogonal polynomials and functions, we may rewrite the representation (1.8) in the form r  r  r  r   r  n 2 n n−1 n n ϕn−1 x − ϕn−2 x ϕn x . (2.20) pn (x) = 2 2 2 2 2 To bound the first term on the right hand side of (2.18), we may use the following lemma. Lemma 2.5. There exists an absolute positive constant C such that 1 pn (0) − ≤ C . π n

Proof. From (2.20) we have p p pn (0) = n/2 ϕ2n−1 (0) − (n − 1)/2 ϕn (0)ϕn−2 (0).

(2.21)

(2.22)

For definiteness we assume that n = 2m. Then ϕ2m−1 (0) = 0,

(−1)m

p (2m)!

ϕ2m (0) = 1 2m m!π 4 p (−1)m−1 (2(m − 1))! ϕ2(m−1) (0) = . 1 2m−1 (m − 1)!π 4

, (2.23)

See for example Gradshtein and Ryzhik ([20], formula (8.953)). The relations (2.22) and (2.23) together imply that for n = 2m p p (2m − 2)! 2m(2m − 1) (2m)!(2m − 2)! √ = 2m−1 √ . pn (0) = 2m−1 (2.24) 2 m!(m − 1)! π 2 ((m − 1)!)2 m π

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Using Stirling’s formula, we get pn (0) =

1 θC + , π n

(2.25)

where θ denotes some quantity, |θ| ≤ 1, thus proving the lemma.



Consider now the second term in (2.18). We put 1 Γn = 2 n

x

Z

√

0

4 − u2

Z

0

u

p′′′ n (s)ds 3

(4 − s2 ) 2

!

du.

(2.26)

Integrating by parts, we get Z u ′′′ pn (s)ds (1) (2) (3) (4) (5) (6) Bn (u) := 3 = Bn (u)+Bn (u)+Bn (u)+Bn (u)+Bn (u)+Bn (u), (2.27) 2 0 (4 − s ) 2 where Bn(1) (u) = Bn(3) (u) =

p′′n (u) 3 2

(4 − u2 ) 3up′n (u)

5

u2 ) 2

(4 − 12pn (0) Bn(5) (u) = , 128

,

Bn(2) (u) = −

,

Bn(4) (u) = −

p′′n (0) , 8 12pn (u)(1 + u2 )

Bn(6) (u) = 60

, 7 (4 − u2 ) 2 Z u pn (s)s(3 + s2 )ds 9

0

(4 − s2 ) 2

.

Once again integrating by parts, we obtain 10 1 X (ν) Γ , Γn = 2 n ν=1 n

(2.28)

where Γ(1) n Γ(3) n Γ(5) n Γ(7) n Γ(9) n

p′n (x) p′n (0) (2) , Γn = − , = (4 − x2 ) 4 Z x 2xpn (x) pn (u)(4 + 3u2 )du (4) =− , Γ = 2 , n (4 − x2 )2 (4 − u2 )3 0 Z p′′n (0) x √ 3xpn (x) =− 4 − u2 du, Γ(6) , n = 8 (4 − x2 )2 0 Z x Z x pn (u)(4 + 3u2)du pn (u)(1 + u2 )du (8) = −3 , Γ = −12 , n (4 − u2)3 (4 − u2 )3 0 0 Z Z x√ Z u 3pn (0) x √ pn (s)(7 + s2 )ds (10) 2 2 = 4 − u du, Γn = 60 4 − u du . 9 32 (4 − s2 ) 2 0 0 0 (3)

(6)

(4)

(7)

(8)

(2)

Collecting Γn , Γn , Γn , Γn , and Γn , and taking into account that Γn = 0, we have (3) (4) (5) (2) Γn = A(1) n + An + An + An + An ,

(2.29)

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where A(1) n A(4) n

Z x p′n (x) xpn (x) pn (s)(16 + 15s2 )ds (2) (3) = 2 , A = , A = − , n n n (4 − x2 ) n2 (4 − x2 )2 n2 (4 − s2 )3 0 Z u Z Z 3pn (0) x √ 60 x √ pn (s)(7 + s2 )ds (5) 2 2 = 4 − u du, A = 4 − u du . 9 n 32n2 0 n2 0 (4 − s2 ) 2 0 2

For some constant γ > 0 we put ε = γn− 3 and introduce an interval Iε = [−2 + ε, 2 − ε]. Note that sup |EFn (x) − G(x)| ≤ sup |EFn (x) − G(x)| + 2G(−2 + ε). (2.30) x

x∈Iε

Lemma 2.6. Assume that there exists some constant C1 such that for any n ≥ 1 1

sup |pn (x) − g(x)| ≤ C1 n− 3 .

(2.31)

sup |EFn (x) − G(x)| ≤ C2 n−1

(2.32)

x∈Iε

Then the following inequality

x∈Iε

holds for some other constant C2 depending on C1 only. Proof. Using (2.18), Lemma 2.5, and the representation (2.29), we obtain 8

∆(ε) n

C X (ν) D , := sup |EFn (x) − G(x)| ≤ + n ν=1 n x∈Iε

(2.33)

where |p′n (x)| , 2 2 x∈Iε n (4 − x )

Dn(1) = sup

Dn(6) Dn(7) Dn(8)

|x|

(2.34) 3 , n2 (4 − x2 ) 2 Z |x| (16 + 15s2 )ds = sup 5 , x∈Iε 0 n2 (4 − s2 ) 2 x∈Iε

|x||pn (x) − g(x)| , Dn(4) n2 (4 − x2 )2 x∈Iε Z |x| (16 + 15s2 )|pn (s) − g(s)| ds = sup , n2 (4 − s2 )3 x∈Iε 0 Z |x| √ 3pn (0) = sup 4 − u2 du, 32n2 x∈Iε 0 Z Z u 60 |x| √ (7 + s2 ) ds 2 = sup 2 4 − u du , (4 − s2 )4 x∈Iε n 0 0 Z Z u 60 |x| √ |pn (s) − g(s)|(7 + s2 ) ds 2 = sup 2 4 − u du . 9 x∈Iε n (4 − s2 ) 2 0 0

Dn(3) = sup Dn(5)

Dn(2) = sup

At first we note that p p p p′n (x) = − n/2ϕn (x n/2)ϕn−1 (x n/2).

(2.35)

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

Using Theorem 8.91.3 from Szeg¨o [28], we get 1

sup |p′n (x)| ≤ Cn 3 .

(2.36)

x∈Iε

This implies that

1 C ≤ . 2 −x ) n

1

Dn(1) ≤ Cn 3 sup x∈Iε

n2 (4

It is easy to see that Dn(2) ≤

(2.37)

C . n

(2.38)

By assumption (2.31), we have |x| C ≤ . 2 2 −x ) n

1

Dn(3) ≤ Cn− 3 sup x∈Iε

(2.39)

n2 (4

Integration with respect to s yields Dn(4) ≤ sup x∈Iε

C n2 (4 − x2 )

Furthermore, 1

Dn(5) ≤ Cn− 3 sup x∈Iε

n2 (4

Similarly, Dn(6) ≤

3 2

≤

C . n

(2.40)

1 C ≤ . 2 2 −x ) n

(2.41)

C . n

(2.42)

Integration with respect to u and s yields Dn(7) ≤ sup x∈Iε

C n2 (4 − x2 )

3 2

≤

C . n

(2.43)

Similarly to (2.39) we get 1

Dn(8) ≤ Cn− 3 sup x∈Iε

n2 (4

C C ≤ . 2 2 −x ) n

(2.44)

Relations (2.33)–(2.44) together imply the result of the lemma.



To investigate pn (x) − g(x) we prove the following lemma. Lemma 2.7. The inequality |p′′n (x)| ≤ ng(x) + n|pn (x) − g(x)| + Cn + √

C 4 − x2

(2.45)

holds for all x ∈ Iε and some positive constant C depending on γ only. p Proof. We introduce the notation xn := x n/2. It is well-known that r n−1 1 X 2 pn (x) = ϕ (xn ). 2n k=0 k

(2.46)

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

677

(See, for example, [26], p. 93). Applying the Christoffel–Darboux formula for orthogonal polynomials (see, for instance, [26], p. 421), we get pn (x) = This equality implies


1 ′ ϕn (xn ) ϕn−1 (xn ) − ϕn (xn ) ϕ′n−1 (xn ) . 2 p′n (x)

and

=

r

n ϕn (xn ) ϕn−1 (xn ) 2

(2.47)

(2.48)


n ϕn (xn ) ϕ′n−1 (xn ) + ϕ′n (xn ) ϕn−1 (xn ) . (2.49) 2 Using the following well-known relation for Hermitean orthogonal functions (see, for example, [20], p. 1057) √ ϕ′n (xn ) = −xn ϕn (xn ) + 2n ϕn−1 (xn ) , p ϕ′n−1 (xn ) = −xn ϕn−1 (xn ) + 2(n − 1) ϕn−2 (xn ) , (2.50) p′′n (x) =

we have

p′′n (x)

r n 2 = −nxn ϕn (xn ) ϕn−1 (xn ) + n ϕ (xn ) 2 n−1 r n−1 +n ϕn (xn ) ϕn−2 (xn ) . 2

Using now the following recurrence formula r  r  n n−1 ϕn−2 (xn ) , ϕn x = xϕn−1 (xn ) − 2 n

(2.51)

(2.52)

we obtain

r
n 2 p′′n (x) = n ϕn−1 (xn ) − ϕ2n (xn ) . 2 Equations (2.47) and (2.50) together imply that r r n 2 n−1 ϕn−1 (xn ) − ϕn (xn ) ϕn−2 (xn ) . pn (x) = 2 2 From relations (2.53) and (2.54) it follows that r n−1 p′′n (x) = npn (x) − n (ϕn (xn ) − ϕn−2 (xn )) ϕn (xn ) 2 r r ! n−1 n +n − ϕ2n (xn ) . 2 2

(2.53)

(2.54)

(2.55)

The table on p. 700 of Askey and Wagner [1] (see also Muckenhaupt ([27], p. 435) shows that the inequalities, for x ∈ Iε , 1

2

1

2

1

|ϕn (xn )| ≤ Cn− 4 (n− 3 + |4 − x2 |)− 4 , 1

|ϕn (xn ) − ϕn−2 (xn )| ≤ Cn− 4 (n− 3 + |4 − x2 |) 4

(2.56)

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

hold for for some positive constant C. Combining (2.55) and (2.56), we get, for x ∈ Iε , |p′′n (x)| ≤ ng(x) + n|pn (x) − g(x)| + Cn + √

C . 4 − x2

(2.57) 

Lemma 2.8. There exist some absolute constants, γ and C, such that 1

sup |pn (x) − g(x)| ≤ Cn− 3 .

(2.58)

x∈Iε

Proof. According to (2.16), (2.27), and Lemma 2.5, we have √  6 X 4 − x2 (ν) C sup |pn (x) − g(x)| ≤ sup |Bn (x)| + . 2 n n x∈Iε ν=1 x∈Iε

(2.59)

Lemma 2.7 implies sup x∈Iε

√

4 − x2 (1) |Bn (x)| n2

It is easy to see that sup x∈Iε

√



≤

C n

1 3

+

C 1

n3

4 − x2 (2) |Bn (x)| n2



sup |pn (x) − g(x)|.

(2.60)

x∈Iε

≤

C 1

n3

.

(2.61)

Furthermore, relations (2.56) and (2.48) together yield |p′n (x)| ≤ √

C . 4 − x2

From this inequality it follows immediately that √  4 − x2 (3) C sup |Bn (x)| ≤ 1 . 2 n x∈Iε n3

(2.62)

(2.63)

For the fourth summand in the right hand side of (2.59) we have the following bound ( ) √  4 − x2 (4) 1 sup |Bn (x)| ≤ C sup 5 n2 x∈Iε x∈Iε n2 (4 − x2 ) 2   1 + C sup sup |pn (x) − g(x)|. (2.64) n2 (4 − x2 )3 x∈Iε x∈Iε Without loss of generality we may assume that   1 1 ≤ . C sup 2 2 3 n (4 − x ) 4 x∈Iε Combining the last two inequalities, we get √  1 4 − x2 (4) C sup |Bn (x)| ≤ 1 + sup |pn (x) − g(x)|. 2 n 4 x∈Iε x∈Iε n3

(2.65)

(2.66)

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

By simple calculations, sup x∈Iε

√

4 − x2 (5) |Bn (x)| n2



≤

C 1

n3

.

An integration shows that ) ( √  4 − x2 (6) 1 sup |Bn (x)| ≤ C sup 5 n2 x∈Iε x∈Iε n2 (4 − x2 ) 2   1 + C sup sup |pn (x) − g(x)|. n2 (4 − x2 )3 x∈Iε x∈Iε

679

(2.67)

(2.68)

This implies that sup x∈Iε

√

4 − x2 (6) |Bn (x)| n2



≤

C n

+

1 3

1 sup |pn (x) − g(x)|. 4 x∈Iε

(2.69)

Relations (2.59), (2.60), (2.61), (2.63), (2.66), (2.67), and (2.69) together imply sup |pn (x) − g(x)| ≤

x∈Iε

C n

1 3

+

3 sup |pn (x) − g(x)|. 4 x∈Iε

The last inequality concludes the proof of lemma.

(2.70) 

2.2 Proof of Theorem 1.1 The results of Lemmas 2.6, 2.8 and inequality (2.30) together complete the proof of Theorem 1.1. 

2.3 Proof of Theorem 1.2 Consider the representation (2.16) of the density pn (x) √ √ Z 4 − x2 4 − x2 x p′′′ n (u)du pn (0) + pn (x) = 3 . 2 2 2 n 0 (4 − u ) 2 Applying Lemma 2.5, similarly to (2.59) we obtain √ √ 6 C 4 − x2 4 − x2 X (ν) |pn (x) − g(x)| ≤ + |Bn (x)|. n n2 ν=1 Applying Lemma 2.7, we get √ 4 − x2 (1) C C C |Bn (x)| ≤ + sup 3 |pn (x) − g(x)| + 5 . 2 2 n n(4 − x ) u∈Iε n(4 − u2 ) 2 n2 (4 − x2 ) 2

(2.71)

(2.72)

If the constant γ is chosen to satisfy sup u∈Iε

C n(4 −

3

u2 ) 2

1 ≤ , 4

(2.73)

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

then inequalities (2.72) and (2.73) together imply √ 4 − x2 (1) 1 C |Bn (x)| ≤ |pn (x) − g(x)| + . 2 n 4 n(4 − x2 ) Furthermore, it is straightforward to check that for x ∈ Iε and ν = 2, 5, √ C 4 − x2 (ν) |Bn (x)| ≤ . 2 n n(4 − x2 ) Applying inequality (2.36), we get √ C C 4 − x2 (3) C |Bn (x)| ≤ 2 ≤ . 2 2 n n(4 − x2 ) n 3 (4 − x2 ) n(4 − x ) Furthermore, for x ∈ Iε the following inequality √ 4 − x2 (4) C C |Bn (x)| ≤ 2 |pn (x) − g(x)| + 5 2 2 3 n n (4 − x ) n2 (4 − x2 ) 2 C 1 ≤ |pn (x) − g(x)| + 4 n(4 − x2 ) holds. From Lemma 2.6 we derive √ 4 − x2 (6) C C C |Bn (x)| ≤ 7 . + 5 ≤ 2 n n(4 − x2 ) n 3 (4 − x2 )3 n2 (4 − x2 ) 2

(2.74)

(2.75)

(2.76)

(2.77)

(2.78)

Inequalities (2.71)–(2.78) together imply that for x ∈ Iε 3 C |pn (x) − g(x)| ≤ |pn (x) − g(x)| + . 4 n(4 − x2 ) The last inequality concludes the proof of the Theorem 1.2.

3

(2.79) 

Sample covariance matrices. The LUE

In this Section we prove Theorems 1.3–1.6. (m) Let pn (x) be the density of the expected spectral distribution function of a matrix (m) W from LUE, and let fn (t) be its characteristic function. Haagerup and Thorbjørnsen ([22], equality (6.19)) proved (fn(m) (t))′

 −(m+n) it =i 1− F (1 − m, 1 − n, 2 ; −t2 /m2 ), m

(3.1)

where F (α, β, γ; s) denotes the hypergeometric function which defined by the following equality F (α, β, γ; s) = 1+

αβ α(α + 1)β(β + 1) 2 α(α + 1)(α + 2)β(β + 1)(β + 2) 3 s+ s + s +· · · . γ · 1! γ(γ + 1) · 2! γ(γ + 1)(γ + 2) · 3!

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

681

The function u(s) = F (α, β, γ; s) satisfies the following differential equation s(1 − s)

du d2 u + [γ − (α + β + 1)s] − αβu = 0 ds2 ds

(3.2)

(see, for example, [20], p. 1072). Taking into account the actual parameters and arguments of the function F , we rewrite this equation as follows:   t2 t2 − 2 1 + 2 F ′′ (1 − m, 1 − n, 2; −t2 /m2 ) m m   t2 + 2 + (3 − m − n) 2 F ′ (1 − m, 1 − n, 2; −t2 /m2 ) m − (1 − m)(1 − n)F (1 − m, 1 − n, 2; −t2 /m2 ) = 0. (3.3) Let y =

n . m

We introduce the notation h(x) := xp(m) n (x),

ϕ(t) = −i(fn(m) (t))′ ,

(3.4)

and prove the following Lemma 3.1. The function h(x) satisfies the following differential equation   a+b xh′′ (x) x2 h′′′ (x) ′ (x − a) (b − x) h (x) + x − h(x) + + = 0, 2 m2 m2 √ √ where a = (1 − y)2 , b = (1 + y)2 .

(3.5)

Proof. Consider the function  m+n it ϕ(t). ψ(t) := 1 − m

(3.6)

ψ(t) = F (1 − m, 1 − n, 2 ; −t2 /m2 ).

(3.7)

According to (3.1), we have

Taking the first and the second derivatives with respect to parameter t in (3.7), we obtain 2t ′ F (1 − m, 1 − n, 2 ; −t2 /m2 ), m2 2 4t2 ψ ′′ (t) = − 2 F ′ (1 − m, 1 − n, 2 ; −t2 /m2 ) + 4 F ′′ (1 − m, 1 − n, 2 ; −t2 /m2 ). m m ψ ′ (t) = −

(3.8)

Solving equations (3.7) and (3.8) with respect to function F (1 − m, 1 − n, 2 ; −t2 /m2 ) and its derivatives, we get m2 ′ ψ (t), 2t   m4 1 ′ ′′ 2 2 ′′ F (1 − m, 1 − n, 2 ; −t /m ) = − 2 ψ (t) − ψ (t) . 4t t F ′ (1 − m, 1 − n, 2 ; −t2 /m2 ) = −

(3.9)

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

We substitute these results to (3.3) and obtain     t2 t2 4(1 − m)(1 − n) ′′ t 1 + 2 ψ (t) + 3 + (5 − 2m − 2n) 2 ψ ′ (t) + t ψ(t) = 0. m m m2 (3.10) Furthermore, taking derivatives in (3.6) yields  m+n−1  m+n it it m+n ′ ψ (t) = −i 1− ϕ(t) + 1 − ϕ′ (t), m m m  m+n−2 it (m + n)(m + n − 1) ′′ 1− ϕ(t) ψ (t) = − m2 m  m+n−1  m+n 2i(m + n) it it ′ − 1− ϕ (t) + 1 − ϕ′′ (t). (3.11) m m m Using these expressions in (3.10), we get
− (1 − y)2 t + 3i(1 + y) ϕ(t) + (3 − 2it(1 + y)) ϕ′ (t) + t ϕ′′ (t) 4t 5t2 t3 + 2 ϕ(t) + 2 ϕ′ (t) + 2 ϕ′′ (t) = 0. m m m

(3.12)

2

Since the function F (1−m; 1−n; − mt 2 ) is polynomial with respect to t of order min{2m, 2n}− 2, according to (3.6) and (3.7), we have ϕ(t)(t) = (|t|−2 ), for |t| → ∞. Applying now the inverse Fourier transform, we get
xh′′ (x) x2 h′′′ (x) −x2 − (1 − y)2 + 2(1 + y)x h′ (x) + (x − (1 + y)) h(x) + + = 0. (3.13) m2 m2

We may rewrite this equality in the following form:   a+b xh′′ (x) x2 h′′′ (x) ′ h(x) + + = 0. (x − a) (b − x) h (x) + x − 2 m2 m2 Inequality (3.14) proves the lemma.

(3.14) 

3.1 Applications of Lemma 3.1 (m)

Remark 3.2. (Integrodifferential equation for pn (x)) Using Lemma 3.1, we obtain the following representation of the function h(x), for a ≤ x ≤ b, p   2 (x − a)(b − x) b+a h(x) = h b−a 2 p Z x (x − a)(b − x) (uh′′ (u) + u2 h′′′ (u)) du − . (3.15) 3 a+b m2 ((u − a)(b − u)) 2 2 We introduce the conjugate differential operator     x3 d3 5x2 d2 4 d a+b (m) Dn (x) := 2 3 + 2 2 − x (x − a)(b − x) + 2 + 3x x − (3.16) m dx m dx m dx 2

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

and a class of functions f : R 7→ R such that n o k (ν) R(m) := f : R → 7 R : lim |x f (x) exp{−mx}| = 0, ν = 0, 1, 2; k ≥ 1 . + n x→∞

683

(3.17)

(m)

Remark 3.3. (Stein type characterizing equation for pn (x)) Let integers n, m satisfy (m) 1 ≤ n ≤ m. A random variable ξ has density pn (x), if and only if, for any function (m) f ∈ Rn ED(m) (3.18) n (ξ)f (ξ) = 0.

3.2 Laguerre polynomial and spectral density of random covariance matrices (q)

Let Lp denote the orthogonal Laguerre function with integer parameters p, q, x

q

(q) − 2 2 (q) L(q) x Lp (x). p = cp e

Here c(q) p =

s

(3.19)

p! , (p + q)!

(3.20)

(q)

and Lp designates a Laguerre polynomial with parameters (p, q). According to Rodrigues’ formula, we have L(q) p (x) =

1 x −q dp −x p+q
e x e x . p! dxp

(3.21)

It is well-known that (see, for example, [22], p. 295) n−1

p(m) n (x)

2 m X  (m−n) = Lj (mx) . n j=0

(3.22)

Using the Christoffel–Darboux formula (see [28], Th. 3.2.2), we get √ m (m) pn (x) = − √ n

(m−n)

dLn dx

! (m−n) dL (m−n) (mx)Ln−1 (mx) − L(m−n) (mx) n−1 (mx) . n dx

(3.23)

We shall use the following relation between the Laguerre polynomials and their derivatives (q)

(q)

(q) pLp (x) − (p + q)Lp−1 (x) dLp (q+1) (x) = −Lp−1 (x) = . dx x

(3.24)

Taking derivative in (3.19) gives (q)

(q)

dLp 1 q (q) − x2 q2 dLp (x) = − L(q) Lp (x) + c(q) x (x). p (x) + p e dx 2 2x dx

(3.25)

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

Combining (3.24) and (3.25), we obtain p (q) (q) (q) pLp (x) − p(p + q)Lp−1 (x) dLp 1 (q) q (q) (x) = − Lp (x) + Lp (x) + . (3.26) dx 2 2x x Applying now the equality (3.26) to the equality (3.23), we obtain √  √  2 m (m−n) (m−n) (m) xpn (x) = √ − Ln−1 (mx)L(m−n) (mx) + mn L (mx) n−1 n n  p (m−n) (mx) . (3.27) − (n − 1)(m − 1)Ln−2 (mx)L(m−n) n Lemma 6.1 from [22] yields √ o d  (m) m m n (m−n) ′ (m−n) h (x) = xpn (x) = √ Ln−1 (mx)Ln (mx) . dx n

(3.28)

Taking the second derivatives in (3.28), we get  5  
d m2 d  (m−n) (m−n) (m−n) (m−n) ′′ h (x) = √ Ln−1 (mx) Ln (mx) + Ln−1 (mx) L (mx) . (3.29) dx n n dx Applying (3.26), we may rewrite the last equation as follows: 3 m2 n (m−n) ′′ xh (x) = √ (m + n − mx − 1) Ln−1 (mx)L(m−n) (mx) n n p (m−n) − (n − 1)(m − 1)Ln−2 (mx)L(m−n) (mx) n o √ (m−n) (m−n) + mnLn−1 (mx)Ln−1 (mx) .

(3.30)

Using the equality

(m−n)

(m−n)

n L(m−n) (mx) = (n + m − 1 − mx) Ln−1 (mx) − (m − 1) Ln−2 (mx), n

(3.31)

(see for instance [20], formula (8.971)), we obtain p √ (m−n) (m−n) (n+m−1−mx)Ln−1 (mx) = nmL(m−n) (mx)+ (m − 1)(n − 1)Ln−2 (mx). (3.32) n Equalities (3.30) and (3.32) together imply  2 
2  (m−n) ′′ 2 (m−n) xh (x) = m Ln (mx) − Ln−1 (mx) .

From equality (3.27) we obtain  2 m (m−n) (m−n) (m−n) 2 (mx) m Ln−1 = mxp(m) n (x) + √ Ln−1 (mx)Ln n r p m (m−n) + (m − 1)(n − 1) L (mx)Ln(m−n) (mx). n n−2

(3.33)

(3.34)

Substituting this representation in (3.33) yields 3

′′

xh (x) =

−mxp(m) n (x)

m 2 (m−n) − √ Ln−1 (mx)L(m−n) (mx) n n (m−n)

+ m2 (L(m−n) (mx) − Ln−2 (mx))L(m−n) (mx) n n p (m − 1)(n − 1) (m−n) √ + m2 ( − 1)Ln−2 (mx)L(m−n) (mx). n mn

(3.35)

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

685

3.3 The case m = n. Proof of Theorem 1.4 (0)

(0)

We consider the case m = n separately. Let Ln (x) = Ln (x), and Ln (x) = Ln (x). In this case the density of the limit distribution is given by r 1 4−x m1 (x) = I[0,4] (x). (3.36) 2π x We prove the following lemma. Lemma 3.4. There exists an absolute positive constant C such that |m1 (2) − p(n) n (2)| ≤

C . n

(3.37)

Proof. We use the formulas of Plancherel-Rotach type for Laguerre polynomials (see [28], Th. 8.22.8, p. 200)     1 3π n − 21 − 14 − 41 exp{−x/2}Ln (x) = (−1) (π sin φ) x n (sin 2φ − 2φ) + sin n + 2 4 o 1 +(nx)− 2 O(1) , where x = (4n + 2) cos2 φ and 0 < ǫ ≤ φ ≤ A simple calculation leads to

π 2

(3.38) p n − ǫ. We put x = 2n and φn = arccos n+1 .

π 6 + + O(n−2 ), 2 4n − 6 π 2 sin 2φn−1 − 2φn−1 = 1 − + + O(n−2 ), 2 4n − 2 π 2 sin 2φn − 2φn = 1 − − + O(n−2 ). 2 4n + 2 The relations (3.38) and (3.39) together imply that     nπ 3π n − 21 − 12 −1 Ln−2(2n) = (−1) (π) n sin n − + + O(n ) , 2 2     nπ n−1 − 21 − 12 −1 Ln−1(2n) = (−1) (π) n sin n − + π + O(n ) , 2    1 1 nπ π  Ln (2n) = (−1)n (π)− 2 n− 2 sin n − + + O(n−1 ) . 2 2 Using there expressions in (3.27), we get sin 2φn−2 − 2φn−2 = 1 −

p(n) n (2) =

1 + O(n−1 ). 2π

(3.39)

(3.40)

(3.41)

The last equality concludes the proof.



For any ε1 > 0 and ε2 > 0, there exist some positive absolute constants C1 and C2 and such that inequality sup |EFn(n) (x) − M1 (x)| ≤ x

sup x∈[ε1 ,4−ε2 ]

1

3

|EFn(n) (x) − M1 (x)| + C1 ε12 + C2 ε22

(3.42)

686

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704 (n)

(n)

(n)

(n)

2

holds. Let Iε = [ε1 , 4 − ε2 ]. We put ε1 = Cn−2 and ε2 = Cn− 3 , and introduce the notation ∆ε = sup |EFn(n) (x) − M1 (x)|. (3.43) x∈Iε

It is easy to see that (n)

(n)

∆n ≤ ∆ε + M1 (ε1 ) + 1 − M1 (4 − ε2 ) ≤ ∆ε + According to (3.15), we have, for x ∈ Iε , r r 4 − x 1 4−x p(n) p(n) n (x) = n (2) − 2 x n x

Z

x

h′′ (u)du 1

3

u 2 (4 − u) 2

2

+

Z

x

C . n

(3.44)

1

h′′′ (u)u 2 du 3

(4 − u) 2

2

!

.

Integrating by parts on the second integral in (3.45), we get r r 4 − x (n) h′′ (x) h′′ (2) 4 − x 1 (n) pn (2) − 2 + + 2 J(x), pn (x) = 2 x n (4 − x) 2n x n where J(x) :=

r

4−x x

Z

x

2h′′ (u)(1 − u)du 5

1

u 2 (4 − u) 2

2

.

(3.45)

(3.46)

(3.47)

Consider J(x). Integrating by parts in the right hand side of (3.47), we obtain r r Z 2h′ (x)(1 − x) h′ (2) 4 − x 4 − x x 2h′ (u)(2u2 − u + 2) J(x) = + + du. 7 3 x(4 − x)2 4 x x u 2 (4 − u) 2 2

(3.48)

Furthermore, we may write EFn(n) (x)

− M1 (x) = =

Z

(n)

ε1

(n)

ε1

Z

p(n) n (u)

0

p(n) n (u)

0

1 + 2 n

Z

x (n)

ε1




− m1 (u) du +


− m1 (u) du +

h′′ (u)du h′′ (2) − 4−u n2

(pn(n) (2) Z

x (n)

ε1

r

Z

x (n)

ε1


p(n) (u) − m (u) du 1 n

− m1 (2))

Z

x

(n)

ε1

4−u 1 du + 2 u n

r

Z

4−u du u

x (n)

J(u)du.

(3.49)

ε1

Let us present the integral of J(x) in the form: Z x J(u)du = B1 + B2 + B3 , (n)

(3.50)

ε1

where Z r 2h′ (u)(1 − u) h′ (2) x 4−u B1 = du, B2 = du, 2 (n) (n) u(4 − u) 4 u ε1 ε1 ! Z x r Z v ′ 4−v 2h (u)(2u2 − u + 2) B3 = du dv. 3 7 (n) v u 2 (4 − u) 2 ε1 2 Z

x

(3.51)

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687

We have the obvious bound

C|h(2)| 1 B2 ≤ . (3.52) 2 n n2 Furthermore, the table on p. 699 of [1] shows that for α ≥ 0 there exists an absolute positive constant C such that for all integers n ≥ 0 |Lαn (x)| ≤ Cxα/2 ν α/2 ,

1 , ν ν ≤x≤ , 2 3ν ≤x≤ , 2

0≤x≤ 1 ν ν 2

≤ Cx−1/4 ν −1/4 , ≤ Cν −1/4 (ν 1/3 + |x − ν|)−1/4 ,

(3.53)

where ν = 4n + 2α + 2. These inequalities imply that for our case, that is α = 0, we have (0) (Ln (x) := Ln (x)) |Ln (nx)| ≤ C,

0≤x≤

1 ≤ x ≤ 2, 4n2 2 2 ≤ x ≤ 4 − n− 3 .

|Ln (nx)| ≤ Cx−1/4 n−1/2 , |Ln (nx)| ≤ Cn−1/2 (n−2/3 + |x − 4|)−1/4 , The relations (3.28) and (3.54) together imply that for |h′ (x)| ≤ p

1 , 4n2

C x(4 − x)

1 4n2

(3.54)

2

≤ x ≤ 4 − n− 3

.

(3.55)

Applying inequality (3.55), we get, for x ∈ Iε ,

 C 1 C  (n) − 1 −3 2 + (4 − x) 2 |B | ≤ (ε ) ≤ . 1 1 n2 n2 n Similarly, for

1 4n2

(3.56)

2

≤ x ≤ 4 − n− 3 , the bound  C 1 C  (n) − 1 −3 2 + (4 − x) 2 |B | ≤ (ε ) ≤ 3 1 n2 n2 n

(3.57)

holds. The inequality (3.54) implies that, for any integer j, j = 0, . . . , n, and any 0 ≤ x ≤ 4n1 2 , |Lj (nx)| ≤ C. (3.58) Inequality (3.58) and relation (3.22) together imply that, for 0 ≤ x ≤

1 , 4n2

pn (x) ≤ Cn. The last inequality yields Z (n) q ε1
(n) (n) (n) pn (u) − m1 (u) du ≤ Cnε1 + C ε1 ≤ Cn−1 . 0

(3.59)

(3.60)

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Integrating by parts, we get Z x ′′ Z x ′ (n) h′ (x) h′ (ε1 ) h (u)du h (u)du = − . + (n) 2 (n) (n) 4−u 4 − x 4 − ε1 ε1 ε1 (4 − u) Applying (3.55) to bound of the right hand side of (3.55) yields Z  C C  (n) − 1 1 x h′′ (u)du (n) − 3 2 2 (ε1 ) + (ε2 ) ≤ . ≤ n2 ε(n) 4 − u n2 n 1 It is easy to see that

1 n2

Z x r C 4 − u ′′ du ≤ 2 . h (2) (n) n u ε1

(3.61)

(3.62)

(3.63)

Relations (3.46), (3.52), (3.57), (3.58), (3.62), (3.63), and Lemma 3.4 together imply ∆ε ≤

C . n

(3.64)

Inequalities (3.64) and (3.44) complete the proof of Theorem 1.4.



3.4 Proof of Theorem 1.3 We recall that

1 p (x − a)(b − x) I[a,b] (x), (3.65) 2πxy √ √ where a = (1 − y)2 , b = (1 + y)2 . We introduce the interval Iε = [a + ε, b − ε] and consider ∆ε = sup EFn(m) (x) − My (x) . (3.66) my (x) =

x∈Iε

− 32

If we take ε = γn

, for some positive constant γ, we get ∆(m) ≤ ∆ε + Cn−1 . n

(3.67)

Note that such a choice of ε implies that for x ∈ Iε p 1 β(x) = (x − a)(b − x) ≥ Cn− 3 .

(3.68)

Furthermore, according to (3.15), we have p(m) n (x)

β(x) β(x) = √ p(m) n (1 + y) − 2x y m2 x

Z

x

uh′′ (u)du 3

1+y

((u − a)(b − u)) 2 ! Z x u2 h′′′ (u)du + . 3 1+y ((u − a)(b − u)) 2

(3.69)

Integrating by parts in the last integral, we obtain p(m) n (x) =

β(x) (m) xh′′ (x) β(x)(1 + y)2 h′′ (1 + y) 1 + − 2 J(x), √ pn (1 + y) − 2 2 √ 2 2x y m β (x) 8m xy y m

(3.70)

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

where

β(x) J(x) = x

Z

x

u((1 − y)2 − 2u2 + (1 + y)u)h′′(u)du 5

((u − a)(b − u)) 2

1+y

689

.

(3.71)

Integrating by parts in the right hand side of (3.71), we get J(x) = β(x) − x

Z

x

1+y



h′ (x)((1 − y)2 − 2x2 + x(1 + y)) h′ (1 + y)(1 + y) β(x) − 3 ((x − a)(b − x))2 x (4y) 2

− 4u4 − 7(1 + y)u3 + (9(1 − y)2 − 4y)u2 − 5(1 + y)(1 − y)2u  −(1 − y)4 h′ (u)

du 7

((u − a)(b − u)) 2

.

(3.72)

Furthermore, we write EFn(m) (x) − My (x) Z a+ε Z x

(m) = pn (u) − my (u) du + p(m) (u) − m (u) du y n 0 a+ε Z a+ε
= p(m) n (u) − my (u) du 0 Z x Z x β(u) du 1 uh′′ (u) du (m) + 2 + (pn (1 + y) − my (1 + y)) u m a+ε β 2 (u) a+ε Z Z x x (1 + y)2h′′ (1 + y) β(u) du 1 + 2 J(u)du. (3.73) − √ 2 8m y y u m a+ε a+ε We represent the last integral in (3.73) in the form Z x J(u)du = B1 − B2 + B3 ,

(3.74)

a+ε

where x

h′ (u)((1 − y)2 − 2u2)du , β 4 (u) a+ε Z h′ (1 + y)(1 + y)((1 − y)2 − 2(1 + y)2) x β(u)du B2 := , 4y 2 u a+ε Z x Z β(u)du u  B3 := − 4z 4 + 10(1 − y)2 z 2 u a+ε a+ε B1 :=

Z

2

3

− 3(1 + y)(1 − y) z − 2z (1 + y) − (1 − y)

4



h′ (z)dz 7

((z − a)(b − z)) 2

. (3.75)

We have the obvious bound |B2 | ≤ C.

(3.76)

Note that, for 0 < a1 ≤ y ≤ a2 < 1 and x ∈ Iε , we get |L(m−n) (mx)| ≤ p n

C nβ(x)

,

(3.77)

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

and (m−n)

1

|L(m−n) (mx) − Ln−2 (mx)| ≤ Cn− 2 n

p β(x).

(3.78)

For the proof of these inequalities see the Appendix. Applying (3.78) to (3.28), we obtain for x ∈ Iε C |h′ (x)| ≤ . (3.79) β(x) This inequality implies that for x ∈ Iε Z x 3 du |B1 | ≤ C (3.80) ≤ Cε− 2 ≤ Cn. 5 a+ε β (u) Similarly, |B2 | ≤ Cn.

(3.81)

|B3 | ≤ Cn.

(3.82)

Applying (3.78) again, we get Furthermore, integrating by parts, we arrive to Z x uh′′ (u)du xh′ (x) (a + ε)h′ (a + ε)) = − β 2 (u) β 2 (x) β 2 (a + ε)) a+ε Z x ′ h (u)(u2 − (1 − y)2 )du − . β 4 (u) a+ε Using inequality (3.78), we derive the bound Z x ′′ uh (u)du ≤ Cn. β 2 (u) a+ε

(3.83)

(3.84)

Relations (3.33) and (3.78) together imply

|h′′ (1 + y)| ≤ Cn.

(3.85)

Applying inequality (3.85), we get Z (1 + y)2h′′ (1 + y) x β(u)du C ≤ . √ 8m2 y y u n a+ε

Relations (3.80)–(3.86) and Lemma A.3 (see Appendix) together provide Z x C (m) sup (pn (u) − my (u))du ≤ . n x∈Iε a+ε From inequality (3.87) it follows in particular that Z a+ε C p(m) . n (u)du ≤ n 0 R a+ε Since 0 my (u)du ≤ Cn , we obtain from (3.73)

∆ε := sup |EFn(m) (x) − My (x)| ≤ x∈Iε

(3.86)

(3.87)

(3.88)

C . n

Inequalities (3.73) and (3.67) together conclude the proof of Theorem 1.3.

(3.89) 

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691

3.5 Proof of Theorem 1.5 From Lemma 3.4 for (3.68) we obtain C C|h′′ (x)| C|h′′ (2)| 1 √ √ + 2 |J(x)|, |p(n) (x) − m (x)| ≤ + + 1 n 2 2 n x n (4 − x) n x n where J(x) =

r

4−x x

Z

2

x

2h′ (u)(2u2 − u + 2) 7

3

u 2 (4 − u) 2

du.

(3.90)

(3.91)

2

The table on the p. 699 of [1] shows that for x ∈ [γn−2 , 4 − γn− 3 ] 1

1

|Ln (mx) − Ln−2 (mx)| ≤ Cn− 2 (x(4 − x)) 4 .

(3.92)

Applying inequalities (3.92) and (3.54) to (3.35), we get C + Cn. |h′′ (x)| ≤ Cnm1 (x) + Cn|p(n) n (x) − m1 (x)| + p x x(4 − x)

(3.93)

The bound (3.93) implies that

C|h′′ (x)| C C ≤ p |p(n) + n (x) − m1 (x)| 2 n (4 − x) n x(4 − x) n(4 − x) C C + 3 + 2 n(4 − x) n (x(4 − x)) 2 1 C C ≤ |p(n) + . n (x) − m1 (x)| + 4 n(4 − x) nx It is easy to see that

C|h′′ (2)| C √ √ ≤ . n2 x n x

(3.94)

(3.95)

Furthermore, we use the representation (3.72) for J(x). According to (3.72), we have for 2 x ∈ [γn−2 , 4 − γn− 3 ] 1 |J(x)| ≤ V1 + V2 + V3 , (3.96) n2 where √ C|h′ (x)| C|h′ (2)| 4 − x √ V1 : = 2 , V2 := , n x(4 − x)2 n2 x r Z C 4 − x x |h′ (u)| V3 : = 2 du . n x 2 u 32 (4 − u) 27 2

Applying representation (3.28) and inequality (3.54), we obtain, for x ∈ [γn−2 , 4 − γn− 3 ], C |h′ (x)| ≤ p . x(4 − x)

(3.97)

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704 2

Using the last inequality yields for x ∈ [γn−2 , 4 − γn− 3 ] V1 ≤

C 3

n2 x 2 (4

− x)

It is simple to check that V2 ≤

5 2

≤

C . nx(4 − x)

(3.98)

C . nx(4 − x)

(3.99)

Applying inequality (3.97) again and integrating, we get V3 ≤

C 3 2

n2 x (4 − x)

5 2

≤

C . nx(4 − x)

(3.100)

The relations (3.96) and (3.98)–(3.100) together conclude the proof of Theorem 1.5.



3.6 Proof of Theorem 1.6 Using representation (3.70), we derive |p(m) n (x) − my (x)| ≤ A1 + A2 + A3 + A4 ,

(3.101)

where β(x) (m) √ |p (1 + y) − my (1 + y)|, 2x y n x|h′′ (x)| A2 := 2 2 , m β (x) β(x)(1 + y)2|h′′ (1 + y)| , A3 := √ 8m2 xy y 1 A4 := 2 |J(x)|. m A1 :=

Applying Lemma A.3, we obtain Cβ(x) . n Using Lemmas A.2 and A.3 in equality (3.35), we get

(3.102)

A1 ≤

|h′′ (x)| ≤ Cnmy (x) + Cn|p(m) n (x) − my (x)| +

C + Cn. β(x)

(3.103)

This inequality implies A2 ≤

C C C 1 (m) (m) + |p (x) − m (x)| ≤ + |p (x) − my (x)|. y n nβ 2 (x) nβ 2 (x) nβ 2 (x) 4 n

(3.104)

It is straightforward to check that A3 ≤

C . nβ 2 (x)

(3.105)

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693

Using inequalities (3.101)–(3.104), we may prove 1

−3 sup |p(m) . n (x) − my (x)| ≤ Cn

(3.106)

x∈Iε

Inequalities (3.103) and (3.106) together imply that A4 ≤

C . nβ 2 (x)

(3.107)

Inequalities (3.102), (3.104), (3.105) and (3.107) together yield |pn(m) (x) − my (x)| ≤

1 C + |p(m) (x) − my (x)|. 2 nβ (x) 4 n

(3.108)

This completes the proof of Theorem 1.6.



Appendix We consider the Laguerre orthogonal function r n! (m−n)/2 −x/2 (m−n) Ln(m−n) (x) := x e Ln (x), m! (n−m)

where Ln

(A.1)

(x) denotes the Laguerre polynomial. By Rodrigues’ formula, L(m−n) (x) = n

By direct check, we show L(m−n) (x) = n Cauchy’s integral formula gives L(m−n) (mx) n

1 x −(m−n) dn −x m
e x e x . n! dxn

(A.2)


1 dn −xz e (1 + z)m |z=0 . n n! dz 1 = 2πi

I

(A.3)

e−mzx (1 + z)m dz. z n+1

(A.4)

Put F (x, z) = −xz + ln (1 + z) − y ln z.

(A.5)

Equalities (A.1) and (A.4) together imply that 1 L(m−n) (mx) = n 2πi Note that

r

n! (mx)(m−n)/2 e−mx/2 m!

I

exp{mF (x, z)}

1 y ∂F (z, x) = −x + − . ∂z 1+z z

dz . z

(A.6)

(A.7)

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

The function F (x, z) for a ≤ x ≤ b has two saddle points z1,2 defined by the equation xz 2 − z(1 − x − y) + y = 0, p (x − a)(b − x) 1−x−y z1,2 = ±i . (A.8) 2x 2x py It is easy to see that |z1,2 |2 = xy . We put z = reiϕ with r = and introduce the x notations g(ϕ) = F (x, reiϕ ),

q(ϕ) = Re{g(ϕ)},

p(ϕ) = Im{g(ϕ)}.

(A.9)

Direct calculations show that

g(ϕ) = −xreiϕ + ln 1 + reiϕ − y ln reiϕ ,
1 q(ϕ) = −xr cos(ϕ) + ln 1 + r 2 + 2r cos ϕ − y ln r, 2 p(ϕ) = −xr sin(ϕ) + ψ − yϕ, where ψ = arccos

1 + r cos ϕ p

1 + r 2 + 2r cos ϕ

!

.

(A.10)

(A.11)

Additionally we assume that x ≤ 1 − y (this assumption holds if x is close to the left edge √ of the spectrum). We write z1,2 = re±iϕ0 with cos ϕ0 = 1−x−y . At first we investigate 2 xy the behavior of function q(ϕ) in some neighborhood of point ϕ0 . Double differentiating yields 2y sin ϕ(cos ϕ − cos ϕ0 ) , 1 + r 2 + 2r cos ϕ 2 cos ϕ − cos ϕ0 2 (1 + r + 2r cos ϕ0 ) q ′′ (ϕ) = 2y cos ϕ − 2y sin ϕ . 1 + r 2 + 2r cos ϕ (1 + r 2 + 2r cos ϕ)2 q ′ (ϕ) =

(A.12)

It is straightforward to check that q ′ (ϕ) = 0 only if ϕ = ϕ0 , and q ′′ (ϕ) < 0 for π/2 ≥ ϕ ≥ ϕ0 . Note that q(ϕ) − q(ϕ0 ) = −xr(cos ϕ − cos ϕ0 ) +

 1 log x(1 + r 2 + 2r cos ϕ) . 2

(A.13)

Since 1 + r 2 + 2r cos ϕ0 = x1 , we represent the argument of the logarithm in the form x(1 + r 2 + 2r cos ϕ) = 1 + α,

(A.14)

α = 2rx(cos ϕ − cos ϕ0 ).

(A.15)

1 1 q(ϕ) − q(ϕ0 ) = − α + log(1 + α). 2 2

(A.16)

where In these notations

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

695

Simple calculations show that p′ (ϕ) = −xr cos ϕ +

r 2 + r cos ϕ 2xr 2 (cos ϕ − cos ϕ0 )(cos ϕ + r) − y = . 1 + r 2 + 2r cos ϕ 1 + r 2 + 2r cos ϕ

(A.17)

Using notation (A.15), we get 1−x+y α + 2xr cos ϕ0 + 2xr 2 =α p (ϕ) = α 2 + 2xα 2 ′

  1 + x2 − x(1 + y) 1+α . (1 − x + y)(1 + xα)

(A.18)

We introduce the notation β(x) := and prove the following Lemma A.1. Assume that (1 − γ≥1

√

p

(x − a)(b − x) √

y)2 ≤ x ≤ (1 +

(A.19)

y)2 and for some positive constant

1

β(x) ≥ γn− 3 .

(A.20)

Then there exists some positive constant C such that

Proof. Note that in the case

|L(m−n) (mx)| ≤ p n 1−

√

C nβ(x)

y ≤x≤1+

√

.

y

(A.21)

(A.22)

there exists some positive constant ρ, ρ > 0, such that |β(x)| ≥ ρ.

(A.23)

In the last case the result may be obtained using the standard steepest descent method. We consider the case √ √ (1 − y)2 ≤ x ≤ 1 − y (A.24) only. The case (1 +

√

y) ≤ x ≤ (1 +

√

y)2

(A.25)

is similar. We consider the representation (A.6). Note that 1−x−y 1 1 − (1 − y) ln x − y ln y. (A.26) 2 2 2 Using Stirling’s formula and the equality n = ym, we get r m−n n! mx A := (mx) 2 exp{− } exp{mRe F (x, z1,2 )} m! 2 n 1 m 1 1 n m mx = (1 + O( ))n 2 exp{− }(2πn) 4 m− 2 exp{ }(2πm)− 4 exp{− } n 2 2 2 m−n m−n m mx my − m(1−y) − my 2 × m 2 x 2 exp{− + + }x y 2 2 2 2 1 1 = (1 + O( ))y 4 . (A.27) n Re F (x, z1,2 ) = −

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F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

Changing the variables in the contour integral by z = r exp{iϕ} and using the last equality, we obtain 1 1 L(m−n) (mx) = (1 + O(n−1))y 4 n π  Z π × Re exp{imp(ϕ0 )} exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))}dϕ .

0

(A.28)

The last representation implies Z π (m−n) 1 Ln exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))}dϕ . (mx) ≤ 2y 4

(A.29)

0

Without loss of generality we may assume that for 0 ≤ ϕ ≤ ϕ0 1 |α| ≤ . 2

(A.30)

This assumption implies that, for 0 ≤ ϕ ≤ ϕ0 , q(α) ≤ − Let ε = √

1 . nβ(x)

α2 . 6

(A.31)

Note that under our assumption 0 ≤ ε ≤ ϕ0 . 1

In fact, since β(x) ≥ n− 3 , we have √

integrals:

I1 = I2 = I3 = I4 =

Z Z

1 nβ(x)

(A.32)

≤ β(x), and β(x) ≤ ϕ0 . We define the following

ϕ0 −ε

exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))}dϕ,

0 ϕ0 +ε ϕ0 −ε π 2

Z

Zϕπ0 +ε π 2

exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))}dϕ, exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))}dϕ,

exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))}dϕ.

At first we find a bound for the integral I4 . Since q ′ (ϕ) < 0, for

π 2

(A.33)

< ϕ ≤ π,

 2 π 1−x−y q(ϕ) − q(ϕ0 ) ≤ q( ) − q(ϕ0 ) ≤ − . 2 2

(A.34)

Inequalities (A.34) implies that √ √  n y(1 − y) |I4 | ≤ C exp − . 2 

(A.35)

F. G¨otze, A. Tikhomirov / Central European Journal of Mathematics 3(4) 2005 666–704

It is easy to see that

C |I2 | ≤ 2ε ≤ p . nβ(x)

Integrating by parts, we get

I1 = I11 − I12 + I13

697

(A.36)

(A.37)

where I11 I12 I13

0 exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))} = , np′ (ϕ) ϕ0 −ε Z p′′ (ϕ) 1 0 =− exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))} ′ dϕ, n ϕ0 −ε [p (ϕ)]2 Z 0 q ′ (ϕ) =− exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))} ′ dϕ. p (ϕ) ϕ0 −ε

By (A.17), we have √ |p′ (0)| ≥ 1 − 2 y,

1 if y < , 4

1 ≥ y. 4

(A.38)

|p′ (ϕ0 − ε)| ≥ Cεβ(x).

(A.39)

√ |p′ (0)| ≥ y y,

if

Furthermore, Relations (A.38) and (A.39) together imply that |I11 | ≤ Using formula p′′ (ϕ) =

d p′ d α , dα d ϕ

1 |I12 | ≤ n

we get

C C ≤p . nεβ(x) nβ(x)

2rx(1−cos ϕ0 )

  nα2 dα C exp − ≤p . 6 α nβ(x) 2rx(cos(ϕ0 −ε)−cos ϕ0 )

Z

(A.40)

From our representation for p′ (ϕ) and q ′ (ϕ) we have   Z 2rx(1−cos ϕ0 ) nα2 C exp − |I13 | ≤ dα ≤ p . 6 nβ(x) 2rx(cos(ϕ0 −ε)−cos ϕ0 )

(A.41)

(A.42)

The estimates (A.40)–(A.42) together imply

Similarly, we get

|I1 | ≤ p |I3 | ≤ p

C nβ(x) C nβ(x)

.

(A.43)

.

(A.44)

Inequalities (A.35), (A.36), (A.43), and (A.44) together conclude the proof.



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We recall that a = (1 −

√

y)2 ,

b = (1 +

√

y)2 .

We prove the following n and m = m(n). Assume that there exist some positive constants Lemma A.2. Let y = m θ1 and θ2 such that 0 < θ1 ≤ y ≤ θ2 < 1. Then there exist positive constant C and γ 2 2 such that for x ∈ [a + γn− 3 , b − γn− 3 ] 1p (m−n) (m−n) (mx) − Ln−2 (mx) ≤ Cn− 2 β(x). (A.45) Ln

Proof. Using the representation (A.6), we get r I n! dz (m−n) (m−n)/2 Ln (mx) = (mx) exp{−mx/2} exp{mF (x, z)} , m! z s r I (m − 1)m n! z 2 dz (m−n) exp{−mx/2} exp{mF (x, z)} . Ln−2 (mx) = (n − 1)n m! (1 + z)2 z

(A.46)

From (A.46) it follows that (m−n)

L(m−n) (mx) − Ln−2 (mx) = B1 + B2 , n

(A.47)

where B1 : =

r

B2 : = ×

I

  I z2 dz n! (m−n)/2 (mx) exp{−mx/2} exp{mF (x, z)} 1 − , 2 m! y(1 + z) z s !r m(m − 1) 1 n! − (mx)(m−n)/2 exp{−mx/2} n(n − 1) y m! exp{mF (x, z)}

z 2 dz . (1 + z)2 z

First we consider the case 1 − x − y > 0. The other case is similar. Note that s m(m − 1) 1 C − ≤ √ . n(n − 1) y n y

Similarly to (A.28) we obtain  Z π 1 1 B1 = y 4 Re exp{imp(ϕ0 )} exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))} π 0    r 2 exp{2iϕ} dϕ (1 + O(n−1)). × 1− y(1 + r exp{iϕ})2

(A.48)

(A.49)

Recall that x|1 + r exp{iϕ}|2 = 1 + α.

(A.50)

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By the definition of the function q(ϕ), we get 2 r exp{2iϕ} exp{m(q(ϕ) − q(ϕ0 ))} ≤ x2 eα exp{(m − 2)(q(ϕ) − q(ϕ0 ))} 2 y(1 + r exp{iϕ})   m−2 2 α = x e exp (−α + log(1 + α)) . (A.51) 2 According to our assumption, √ √ √ √ 1 0 < 2 y(1 − y) − Cβ 2 (x) < 1 − x − y < 2 y(1 − y) ≤ . 2 Since q(ϕ) is monotonously decreasing, we have, for

π 2

(A.52)

≤ ϕ ≤ π,

 2 1−x−y 1 1−x−y q(ϕ) − q(ϕ0 ) ≤ q(π/2) − q(ϕ0 ) ≤ + log(x + y) ≤ − . (A.53) 2 2 2 This implies that, for π2 ≤ ϕ ≤ π,   2 r exp{2iϕ} γ(m − 2) 2 exp{m(q(ϕ) − q(ϕ0 ))} ≤ Cx exp − , y(1 + r exp{iϕ})2 2

(A.54)

where γ is some positive constant depending on a1 and a2 . Similarly to (A.33) we introduce the representation  1 −1 B1 = 1 + O(n ) (I1 + I2 + I3 + I4 ) , (A.55) π

where

I1 =

Z

ϕ0 −ε

Ψ(ϕ)dϕ,

I2 =

0

I3 =

Z

π 2

Ψ(ϕ)dϕ,

I4 =

ϕ0 +ε

Z

Z

ϕ0 +ε

Ψ(ϕ)dϕ,

ϕ0 −ε π

Ψ(ϕ)dϕ,

(A.56)

π 2

and  Ψ(ϕ) = exp{m(q(ϕ) − q(ϕ0 )) + im(p(ϕ) − p(ϕ0 ))} 1 −

r 2 exp{2iϕ} y(1 + r exp{iϕ})2



.

(A.57)

Using inequalities (A.51) and (A.54), we get |I4 | ≤ C exp{−γn}.

(A.58)

Simple calculations yield 1− Note that

r 2 exp{2iϕ0 } β 2 (x) (1 + y − x)β(x) = +i . 2 y(1 + r exp{iϕ0 }) 2y 2y 2 r 2 exp{2iϕ0 } β 2 (x) 1 − = , y(1 + r exp{iϕ0 })2 y

(A.59)

(A.60)

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and r 2 exp{2iϕ} r 2 exp{2iϕ0 } 1 − − 1− y(1 + r exp{iϕ})2 y(1 + r exp{iϕ0 })2 r 2 exp{2iϕ} r 2 exp{2iϕ0 } ≤ − . y(1 + r exp{iϕ})2 y(1 + r exp{iϕ0 })2

(A.61)

|1 + r exp{iϕ}|2 ≥ 1 + r 2 ≥ 1.

(A.62)

Furthermore, for 0 ≤ ϕ ≤ ϕ0 + ε,

The last two inequalities imply that, for 0 ≤ ϕ ≤ ϕ0 + ε, 2 2 r exp{2iϕ } r exp{2iϕ} 0 1 − − 1 − ≤ Cε ≤ Cβ(x). y(1 + r exp{iϕ})2 y(1 + r exp{iϕ0 })2

Using (A.60), we get, for 0 ≤ ϕ ≤ ϕ0 + ε, r 2 exp{2iϕ} 1 − ≤ Cβ(x). y(1 + r exp{iϕ})2

(A.63)

(A.64)

Inequality (A.64) implies

|I2 | ≤ We define the notation

Cβ(x) √ . n

r 2 exp{2iϕ} κ(x, ϕ) := 1 − , y(1 + r exp{iϕ})2

(A.65)

(A.66)

and similarly to (A.37) we represent the integral I1 in the form I1 = I11 − I12 + I13 + I14 ,

(A.67)

where I11 I12 I13 I14

ϕ −ε exp{mP (ϕ)} 0 , = κ(x, ϕ) np′ (ϕ) 0 Z 1 ϕ0 −ε p′′ (ϕ) =− κ(x, ϕ) exp{mP (ϕ)} ′ dϕ, n 0 [p (ϕ)]2 Z ϕ0 −ε q ′ (ϕ) =− κ(x, ϕ) exp{mP (ϕ)} ′ dϕ, p (ϕ) 0 Z ϕ0 −ε exp{mP (ϕ)} =− κ′ (x, ϕ) dϕ. np′ (ϕ) 0

Here P (ϕ) = (q(ϕ) − q(ϕ0 )) + i(p(ϕ) − p(ϕ0 )). Using inequalities (A.40) and (A.64), we obtain p C β(x) √ |I11 | ≤ . n

(A.68)

(A.69)

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Applying inequality (A.64) once again, we get p C β(x) √ |I12 | ≤ , n and

p

β(x) √ . n It is straightforward to check that, for 0 ≤ ϕ ≤ π2 , |I13 | ≤

C

|κ′ (x, ϕ)| ≤ C.

701

(A.70)

(A.71)

(A.72)

Note that under our assumption 1

3

n− 2 ≤ Cβ(x) 2 .

(A.73)

Using the representation (A.18) and changing variables in the integral I14 , we get p C β(x) √ |I14 | ≤ . (A.74) n Inequalities (A.69), (A.70), (A.71), and (A.74) together imply p C β(x) √ . |I1 | ≤ n

(A.75)

To bound the integral I3 we may again use integration by parts. After simple calculations we obtain p C β(x) √ |I3 | ≤ . (A.76) n It is easy to show p C β(x) √ |B2 | ≤ . (A.77) n Inequalities (A.58), (A.65), (A.75), (A.76) and (A.77) together conclude the proof of the lemma.  Lemma A.3. There exists a positive constant C such that (m) p (1 + y) − my (1 + y) ≤ C . n n

(A.78)

Proof. We give an outline of the proof only. We apply the method of steepest descent √ to the representation (A.28). Since β(1 + y) = 2 y does not depend on n, we obtain s   √ 1 π ϕ0 − ψ0 (m) mLn (m(1 + y)) = √ cos n Im [F (r exp{iϕ0 }, 1 + y)] + − π y 4 2

+O(n−1), (A.79)

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√ where ψ0 = arccos( x(1 + r cos ϕ0 )). Similarly, we get s   √ 1 π ϕ0 − ψ0 (m) mLn−1 (m(1 + y)) = √ cos n Im [F (r exp{iϕ0 }, 1 + y)] + + π y 4 2

+O(n−1), (A.80)

and √

(m) mLn−2 (m(1

+ y)) =

s

n π 1 √ cos n Im [F (r exp{iϕ0 }, 1 + y)] + π y 4  3(ϕ0 − ψ0 ) + + O(n−1 ). 2

(A.81)

Substituting these expressions to (3.27) and using the simple relation, cos2 ((α + β)/2) − cos α cos β = sin2 ((α − β)/2), we get 1 2 −1 (1 + y)p(m) n (1 + y) = √ sin (ϕ0 − ψ0 ) + O(n ). π y

(A.82)

It is straightforward to check that sin(ϕ0 − ψ0 ) =

p

1 + y sin ϕ0 =

√

1 + y β(1 + y) √ = 1. √ 2 1+y y

(A.83)

The last two relations together imply 1 −1 −1 (1 + y)p(m) n (1 + y) = √ + O(n ) = (1 + y)my (1 + y) + O(n ). π y This completes the proof.

(A.84) 

References [1] R. Askey and S. Wainger: “Mean convergence of expansion in Laguerre and Hermitean series”, American Journals of Mathematics, Vol. 87, (1965), pp. 695–707. [2] Z.D. Bai: “Convergence rate of expected spectral distributions of large random matrices. Part I. Wigner matrices”, Ann. Probab., Vol. 21, (1993), pp. 625–648. [3] Z.D. Bai: “Convergence rate of expected spectral distributions of large random matrices. II. Sample covariance matrices”, Ann. Probab., Vol. 21, (1993), pp. 649– 672. [4] Z.D. Bai: “Methodologies in spectral analysis of large dimensional random matrices: a review”, Statistica Sinica, Vol. 9, (1999), pp. 611–661. [5] Z.D. Bai, B. Miao and J. Tsay: “Convergence rates of the spectral distributions of large Wigner matrices”, Int. Math. J., Vol. 1, (2002), pp. 65–90. [6] Z.D. Bai, B. Miao and J.-F. Yao: “Convergence rate of spectral distributions of large sample covariance matrices”, SIAM J. Matrix Anal. Appl., Vol. 25, (2003), pp. 105– 127.

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[7] P. Deift: Orthogonal Polynomials and Random Matrices: A Rieman-Hilbert Approach, Courant Lectures Notes, Vol. 3, Amer. Math. Soc., 2000. [8] P. Deift, T. Kriecherbauer, K.D.T.-R. McLaughlin, S. Venakides and X. Zhou: “Strong asymptotics of orthogonal polynomials with respect to exponential weights”, Comm. Pure and Applied Math., Vol. LII, (1999), pp. 1491–1552. [9] N.M. Ercolani and K.D.T.-R. McLaughlin: “Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques, and applications to grafical enumeration”, Int. Math. Res. Not., Vol. 14, (2003), pp. 755–820. [10] A. Erdelyi: “Asymptotic solutions of differencial equations with transition points or singularities”, J. Math. Phys., Vol. 1, (1960), pp. 16–26. [11] P. Forrester: Log-gases and Random Matrices, Book Manuscript: www.ms.unimelb.edu.au/˜matpjf/matpjf.html [12] V.L. Girko: “Asymptotics distribution of the spectrum of random matrices”, Russian Math. Surveys., Vol. 44, (1989), pp. 3–36. [13] V.L. Girko: “Convergence rate of the expected spectral functions of symmetric 1 random matrices equals to O(n− 2 )”, Random Oper. and Stoch. Equ., Vol. 6, (1998), pp. 359–406. [14] V.L. Girko: “Extended proof of the statement: Convergence rate of the expected 1 spectral functions of symmetric random matrices Ξn is equal to O(n− 2 ) and the method of critical steepest descent”, Random Oper. and Stoch. Equ., Vol. 10, (2002), pp. 253–300. [15] N.R. Goodman: “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)”, Ann. Math. Statistics, Vol. 34, (1963), pp. 152–177. [16] F. G¨otze and A.N. Tikhomirov: “Rate of convergence to the semi-circular law for the Gaussian Unitary Ensemble”, Theory Probab. Appl., Vol. 47, (2002), pp. 381–388. [17] F. G¨otze and A.N. Tikhomirov: “Rate of convergence to the semi-circular law”, Probab. Theory Relat. Fields, Vol. 127, (2003), pp. 228–276. [18] F. G¨otze and A.N. Tikhomirov: “Rate of Convergence in Probability to the Marchenko-Pastur Law”, Bernoulli, Vol. 10(1), (2004), 1–46. [19] F. G¨otze and A.N. Tikhomirov: Limit theorems for spectra of random matrices with martingale structure, Bielefeld University, Preprint 03-018 2003, www.mathemathik.uni-bielefeld.de/fgweb/preserv.html [20] I.S. Gradstein and I.M. Ryzhik: Table of Integrals, Series, and Products, Academic Press, Inc. New York, 1994. [21] J. Gustavsson: Gaussian fluctuations of eigenvalues in the GUE, arXiv: math.PR/0401076 v1, 1–27, (2004). [22] U. Haagerup and S. Thorbjørnsen: Random matrices with complex Gaussian entries, Expo. Math., Vol. 21, (2003), pp. 293–337. [23] R. Janik and M. Nowak: “Wishart and anti-Wishart random matrices”, J. of Phys. A: Math. Gen., Vol. 36, (2003), pp. 3629–3637. [24] M. Ledoux: Differential operators and spectral distribution functions of invariant ensembles from the classical orthogonal polynomials. The continuous case, Preprint, University of Toulouse, 2002, pp. 1–31.

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[25] V.M. Marchenko and L.A. Pastur: “The eigenvalue distribution in some ensembles of random matrices”, Math.USSR Sbornik, Vol. 1, (1967), pp. 457–483. [26] M.L. Mehta: Random matrices, 2nd ed., Academic Press, San Diego, 1991. [27] B. Muckenhaupt: “Mean convergence of Hermitian and Laguerre series I, II”, Trans. American Math. Soc., Vol. 147, (1970), pp. 419–460. [28] G. Szeg¨o: Orthogonal Polynomials, American Math. Soc., New York, 1967.

CEJM 3(4) 2005 705–717

The geometry of Kato Grassmannians Bogdan Bojarski1, Giorgi Khimshiashvili2∗ 1

Institute of Mathematics, Polish Academy of Sciences, Sniadeckich str. 8, Warsaw, Poland 2 A. Razmadze Mathematical Institute, Georgian Academy of Sciences, M. Aleksidze str. 1, Tbilisi 0193, Georgia

Received 7 November 2004; accepted 26 July 2005 Abstract: We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold. c Central European Science Journals. All rights reserved.

Keywords: Fredholm pair of subspaces, Fredholm operator, index, compact operator, restricted Grassmannian, Hilbert manifold, homotopy groups, Fredholm structure MSC (2000): 35E15, 58D15

Introduction We present several basic results about Fredholm pairs of subspaces introduced by T.Kato [15] and discuss global geometric and topological properties of associated Grassmannians, sometimes called Kato Grassmannians or restricted Grassmannians, which have attracted considerable interest in the last two decades (see, e.g., [3-6, 7, 13, 17, 21]). As was shown in [3-6], many geometric aspects of classical linear conjugation problems with sufficiently regular (differentiable, H¨older) coefficients can be formulated and successfully studied in the framework of Fredholm pairs of subspaces and Kato Grassmannians in a complex Hilbert space. In general the concept of Fredholm pair makes sense in an arbitrary Banach space ∗

E-mail: [email protected]

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(see, e.g., [15]). However, Fredholm pairs in a Hilbert space lead to an especially rich theory with many connections to other fields. Specifically, one can define a whole chain of restricted Grassmannians which are related to various topics in the theory of loop groups and mathematical physics [21]. These Grassmannians appear to have interesting geometric properties and can be treated from various points of view. We are basically interested in global geometric properties of those Grassmanians. In particular, we describe their topological structure and show that they can be studied using the so-called Fredholm structures [11]. The main result yields that the restricted Grassmannian can be endowed with a natural Fredholm structure (Theorem 3). Let us add a few words about the structure of the paper. We begin by presenting the most essential geometric and topological properties of Fredholm pairs of subspaces in a Hilbert space. The discussion in this section closely follows [3] and [7] (cf. also [1]). In the second section we give some relevant results on the geometry of Kato Grassmannians and restricted Grassmannians. In conclusion we present an explicit construction of Fredholm structures on restricted Grassmannians.

1

Fredholm pairs and Kato Grassmannians

In the sequel we freely use standard concepts and results of functional analysis and operator theory concerned with bounded linear operators, projections, Fredholm operators and their indices. All Hilbert spaces are supposed to be real or complex and all subspaces are supposed to be closed. We begin by recalling the general concept of a Fredholm pair of subspaces. Consider two (closed) subspaces L1 , L2 of a Banach space E over field K which can be either R or C. Definition 1.1. [15] Pair (L1 , L2 ) is called a Fredholm pair of subspaces (FPS) if their intersection L1 ∩ L2 is finite dimensional and their sum L1 + L2 has finite codimension in E. Then the index of pair (L1 , L2 ) is defined as ind (L1 , L2 ) = dimK (L1 ∩ L2 ) − codimK (L1 + L2 ).

(1)

This concept was introduced by T.Kato who, in particular, proved that the index is invariant under homotopies [15]. To distinguish this concept from similar ones, we sometimes speak of a Kato Fredholm pair. Some modifications of the above definition are presented below and when no confusion is possible they all are referred to as Fredholm pairs. The set of all Kato Fredholm pairs is denoted by F p(E) and called the Kato biGrassmannian of E. In many geometric problems it becomes necessary to consider the set of all Fredholm pairs in E with a fixed first subspace. Such sets are most interesting when the subspace in question is of infinite dimension and infinite codimension (idic). The following definition is implicitly present in [3]. Definition 1.2. (cf. [3]) Let L be a closed infinite dimensional and infinite codimensional

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707

subspace of E. The Kato Grassmannian GrK (L, E) is defined as the set of all closed subspaces M ⊂ E such that (L, M) is a Kato Fredholm pair. This is actually just a ”leaf” in the Kato biGrassmanian F p(E) and in many cases one can represent F p(E) as a fibration over the set of all closed subspaces Gr(E) with the fiber homeomorphic to GrK (L, E). Thus the geometry and topology of F p(E) can be often understood by investigating the Kato Grassmannian and we will follow this strategy in the sequel. As is well known, Fredholm pairs appear in a number of important problems of analysis and operator theory [15, 4, 5, 9, 21]. Global geometric and topological properties of the set of Fredholm pairs play essential role in some recent papers on differential equations and infinite dimensional Morse theory in the spirit of the Floer approach (see, e.g., [1]). Taking this into account it seems remarkable that one can relate the theory of Fredholm pairs with some topics of global analysis in the spirit of Fredholm structures theory [11, 12]. In line with that idea we present in the last section a natural construction of Fredholm structures on certain subsets of Fredholm pairs with a fixed first component. The theory of Fredholm pairs is especially rich and fruitful in the case of a Hilbert space. Therefore we only consider the case where E is a separable Hilbert space H. There are three important peculiarities in the case of a Hilbert space which are worthy of mentioning from the very beginning. First of all, it is well known that the group of unitary operators U(H) transitively acts on the set of idic subspaces (see, e.g., [11]). This trivially implies that, for every two idic subspaces L1 , L2 of a Hilbert space H, the Kato Grassmannians GrK (L1 , H) and GrK (L2 , H) are homeomorphic. For this reason in topological questions one can write simply GrK (H) and speak of Kato Grassmannian of H. Next, in a Hilbert space one can pass to the orthogonal complements of subspaces ⊥ considered. Thus, for each pair of subspaces (L1 , L2 ), one has a dual pair P ⊥ = (L⊥ 1 , L2 ) and it is easy to see that if one of them is a Fredholm pair then the second one also has this property. Since ⊥ ind (L1 , L2 ) = dimK (L1 ∩ L2 ) − codK (L1 + L2 ) = dimK (L1 ∩ L2 ) − dimK (L⊥ 1 ∩ L2 ),

we also have ⊥ ind (L1 , L2 ) = −ind (L⊥ 1 , L2 ).

(2)

Thus there exists a sort of duality for Fredholm pairs in Hilbert spaces. ⊥ Proposition 1.3. A pair P = (L1 , L2 ) is a FPS if and only if P ⊥ = (L⊥ 1 , L2 ) is a FPS and ind P = −ind P ⊥ .

This result means that the operation of passing to orthogonal complements acts on the Kato biGrassmannian. As is explained below, the set of Fredholm pairs can be endowed with a natural structure of infinite dimensional manifold and the above map becomes a smooth involution of F p(H).

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The third peculiarity is that in the setting of Hilbert spaces one can equivalently deal with orthogonal projections on subspaces considered. This is convenient in many constructions and appropriate for generalizations in the context of Banach algebras (cf. [7, 10]). To make this idea more precise we now present a counterpart of Definition 1 in the language of projections. Definition 1.4. ([2]) Let P1 and P2 be orthogonal projections in a separable Hilbert space. A pair (P1 , P2 ) is called a Fredholm pair of projections (FPP) if the operator C12 = P2 P1 |im P1 considered as an operator from im P1 to im P2 is Fredholm, and if this is the case then the index ind (P1 , P2 ) is defined as the Fredholm index of C = C12 . It is remarkable that the two definitions are equivalent as is shown by the following simple but conceptually important proposition which is often used for constructing Fredholm pairs. Proposition 1.5. A pair of orthogonal projections (P1 , P2 ) is a FPP if and only if (im P1 , im P2⊥ ) is a FPS. In such case ind (P1 , P2 ) = ind (imP1 , imP2⊥ ).

(3)

Proof. First of all, putting L1 = imP1 , L2 = imP2 it is easy to verify that ⊥ ⊥ ker C = L1 ∩ L⊥ 2 , (im C) = L1 ∩ L2 . ⊥ ⊥ ⊥ Thus fredholmness of C is equivalent to the fact that L1 ∩ L⊥ 2 and L1 ∩ L2 = (L1 + L2 ) are finite-dimensional and L1 + L⊥ 2 is closed, which is in turn equivalent to the fact that ⊥ (L1 , L2 ) is a FPS. Moreover, ⊥ ⊥ ⊥ ⊥ ind (P1 , P2 ) = dimK (L1 ∩ L⊥ 2 ) − dimK (L1 ∩ L2 ) = dimK (L1 ∩ L2 ) − dimK (L1 + L2 ) = ⊥ ⊥ dimK (L1 ∩ L⊥ 2 ) − codimK (L1 + L2 ) = ind(L1 , L2 ),

which finishes the proof. Thus we have two equivalent concepts of a Fredholm pair each of which has its specific applications and suggests further generalizations. There also exist more restrictive notions of Fredholm pair which appear useful in operator theory and functional analysis. We describe here one of them which plays an important role in the study of linear conjugation problems and Grassmannian embeddings of loop groups. It is based on the concept of commensurability of projections which can be easily generalized to Banach algebra context. Definition 1.6. (cf. [1]) Let P1 , P2 be orthogonal projections in a separable Hilbert space. They are called commensurable if their difference P1 − P2 is a compact operator and then it is also said that (P1 , P2 ) is a commensurable pair of projections (CPP). A pair of subspaces (L1 , L2 ) is called a commensurable pair of subspaces (CPS) if orthogonal projectors on these subspaces are commensurable.

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Proposition 1.7. Two subspaces L1 , L2 are commensurable if and only if the operators PL⊥1 PL2 and PL⊥2 PL1 are compact. Proof. The ”only if” implication is trivial. Indeed, we have that PL2 − PL1 = K is a compact operator. Multiplying this equality on the left by PL⊥1 and taking into account that PL⊥1 PL1 = 0 we get PL⊥1 PL2 = PL⊥1 K which is obviously compact. The ”if” implication follows by another easy calculation: P1 − P2 = (P2 + P2⊥ )P1 − P2 (P1 + P1⊥ ) = P2 P1 + P2⊥ P1 − P2 P1 − P2 P1⊥ = P2⊥ P1 − P2 P1⊥ , where we used the evident fact that, for each orthogonal projection P , one has I = P +P ⊥ .  It is also known that a pair of subspaces (L1 , L2 ) is Fredholm if L1 and L⊥ 2 are commensurable but the converse is not true (see, e.g., [1]). Notice that the notion of commensurability makes sense for an arbitrary Banach space. This notion enables one to introduce a subset of the Kato Grassmannian playing an important role in many problems of functional analysis. Definition 1.8. Let L be an idic subspace of H. The set of all subspaces commensurable with L is called the commensurable Grassmannian Grc (L, H). For a subspace M ∈ Grc (L, H), the relative dimension of L with respect to M is defined as the integer dim(L, M) = ind(L, M ⊥ ). From the above discussion it easily follows that the relative dimension is constant on connected components of Grc (L, H). In fact, it distinguishes the components. Thus each component consists of subspaces M commensurable with L and such that dim(L, M) is equal to a fixed integer. It is easy to see that the topological type of Grc (L, H) is the same for all idic subspaces L so one can simply write Grc (H) and speak of commensurable Grassmannian of H (it is also called the Grassmannian of compact perturbations [1]). It can be obviously identified with a subspace of the Kato Grassmannian GrK (H). In order to investigate the geometry of Fredholm pairs and Kato Grassmannians more closely it appears useful to consider certain smaller Grassmannians which will be our main concern in the next section. To conclude this section we want to point out that one can give a useful parameterization of Kato Fredholm pairs in terms of an associated operator group which is a sort of analog of the classical singular integral operators and appears useful in the geometric study of linear conjugation problems [3] (cf. [18]). In particular, one can construct the subspaces from GrK (L, H) using the following construction suggested in [3]. This construction is only interesting when dim L = dim L⊥ = ∞, so we suppose that this is the case. For a given operator A ∈ B(H) denote by A′e the essential commutant of A, i.e., the set of all operators T such that the commutator [T, A] is compact. If P is an orthogonal projection in a separable Hilbert space such that dim im P = dim ker P = ∞, then the

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algebra Pe′ can be related to the classical singular integral operators on a closed contour by taking in the role of P the Szeg¨o projector on the Hardy space in L2 (S 1 , Cn ) [3]. Invertible operators from Pe′ can be used for constructing Kato Fredholm pairs. Denote by Ge (P ) the set of all invertible operators from Pe′ . Fix now a subspace L, denote by P the orthogonal projection on L and put Q = I − P . The following result which was established in [3] can be proven similarly to Proposition 1.7. Proposition 1.9. ([3]) For each A ∈ Ge (P ), the pair (L, AL⊥ ) is a Fredholm pair of subspaces. The operator Φ = P + AQ is a Fredholm operator and ind Φ = ind (L, AL⊥ ). Analogously, one can parameterize the commensurable Grassmannian using another operator group. Denote by GLc (H) the group of linear automorphisms of H which are compact perturbations of the identity. It is elementary to verify that, for any projection P and any A ∈ GLc (H), the projection AP is commensurable with P . Thus GLc (H) acts on the commensurable Grassmannian and it is easy to see that this action is analytic and preserves the connected components of Grc (L, H). Proposition 1.10. (cf. [1]) The group GLc (H) analytically acts on Grc (L, H), the relative dimension is invariant under this action, and the action is transitive on each connected component of Grc (L, H). We do not give the proof of the latter proposition because more precise results will be presented in the next section in terms of the so-called restricted Grassmannians.

2

Kato Grassmannians and restricted Grassmannians

The concepts of Kato Grassmannian and commensurable Grassmannian permit useful modifications which we present following [21] and [13]. It is convenient to introduce the Grassmannian Gr(H) defined as the collection of all closed subspaces in H. The assignment L 7→ PL defines an inclusion of Gr(H) into B(H) with the image equal to the set of all orthogonal projections in H. One can now define the metric on Gr(H) by putting the distance between two subspaces to be the norm of the difference of orthogonal projections on these subspaces. This makes Gr(H) into a complete metric space and it can be proved that it has a natural structure of an analytic Banach submanifold induced from B(H) [1]. The Kato biGrassmannian F p = F p(H) is an open subset of Gr(H) × Gr(H) so we endow it with the induced topology and then the index is a continuous function on F p. Denote by F p∗ the subset consisting of Fredholm pairs consisting of infinite dimensional subspaces. The connected components of F p∗ are exactly the subsets F p∗k consisting of Fredholm pairs with index k and it can be proved that all components are homeomorphic (see, e.g., [21]). For a real Hilbert space, each of them has the homotopy type of the classifying space of infinite orthogonal group BO(∞) and so its homotopy groups are

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well known. Thus F p∗ is homotopy equivalent to Z × BO(∞), its homotopy groups are 8-periodic by Bott periodicity, and the first 8 homotopy groups are: Z, Z2 , Z2 , 0, Z, 0, 0, 0. In the complex case one has the same picture with 2-periodic homotopy groups beginning with Z, 0. We are now prepared to introduce and investigate the so-called restricted Grassmannians. Consider a complex Hilbert space decomposed into an orthogonal direct sum H = H+ ⊕ H− and choose a positive number s. For further use we need a family of subideals in the ideal of compact operators K(H) which is defined as follows (cf. [13]). Recall that for any bounded operator A ∈ L(H) the product A∗ A is a non-negative self-adjoint operator, so it has a well-defined square root |A| = (A∗ A)1/2 (see, e.g., [11]). If A is compact, then A∗ A is also compact and |A| has a discrete sequence of eigenvalues µ1 (A) ≥ µ2 (A) ≥ . . . tending to zero. The µn (A) are called singular values of A. For a finite s ≥ 1 one can consider the expression (sth norm of A) ||A||s =

"∞ X j=1

(µj (A))s

#1/s

(4)

and define the sth Schatten ideal Ks as the collection of all compact operators A with a finite sth norm (s-summable operators) [11]. Using elementary inequalities it is easy to check that Ks is really a two-sided ideal in L(H). These ideals are not closed in L(H) with its usual norm topology but if one endows Ks with the sth norm as above then Ks becomes a Banach space [11]. Two special cases are well-known: K1 is the ideal of trace class operators and K2 is the ideal of Hilbert-Schmidt operators. For s = 2, the above norm is called the Hilbert-Schmidt norm of A and it is well known that K2 (H) endowed with this norm becomes a Hilbert space (see, e.g., [11]). Obviously K1 ⊂ Ks ⊂ Kr for 1 < s < r so one obtains a chain of ideals starting with K1 . For convenience we set K∞ = K and obtain an increasing chain of ideals Ks with s ∈ [1, ∞]. Of course one can introduce similar definitions for a linear operator A acting between two different Hilbert spaces, e.g., for an operator from one subspace M to another subspace N of a fixed Hilbert space H. In particular we can consider the classes Ks (H± , H∓ ). Let us also denote by F (M, N) the space of all Fredholm operators from M to N. Definition 2.1. ([21]) The sth restricted Grassmannian of a polarized Hilbert space H is defined as Grrs (H) = {W ⊂ H : π+ |W ∈ F (W, H+ ), π− |W ∈ Ks (W, H− )}. In general, two subspaces are called s-commensurable if the difference of orthogonal projections on these subspaces belongs to Ks . Thus the sth Grassmannian is the collection of all subspaces s-commensurable with H+ . These Grassmannians are of major interest

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to us. Actually, many of their topological properties (e.g., the homotopy type) do not depend on the number s appearing in the definition. On the other hand, more subtle geometric properties like manifold structures and characteristic classes of GrFs do depend on s in a quite essential way. As follows from the discussion in [13] this is a delicate issue and we circumvent it by properly choosing the context. As follows from the results of [21], it is especially convenient to work with the Grassmannian Grr2 (H) defined by the condition that the second projection π− restricted to W is a Hilbert-Schmidt operator. Following [21] we denote it by Grr (H) and call it the restricted Grassmannian of H. The restricted Grassmannians have interesting analytic and topological properties similar to those of Kato Grassmannians and commensurable Grassmannians. It can be shown that the Grassmannian Grrs can be turned into a Banach manifold modeled on Schatten ideal Ks . In particular Grr (H) has a natural structure of a Hilbert manifold modeled on the Hilbert space K2 (H) [21]. All these Grassmannians have the same homotopy type (see Theorem 2.6 below). Moreover, as we will see in the last section, Grassmannians Grrs can be endowed with natural Fredholm structures [11], which in particular suggests that one can define various global topological invariants of Grrs (H). By analogy with Definition 2.1 one can introduce a family of subgroups GLsr of GL(H) (s ≥ 1). Definition 2.2. ([21]) The subgroup GLsr (H) is defined as the set of all invertible bounded operators A such that the commutator [A, π+ ] belongs to the Schatten class Ks (H). The subgroup GLr = GL2r , called the restricted linear group of H, is especially important for our considerations. From the very definition it follows that GLsr acts on Grrs and it was shown in [7] that these actions are transitive (cf. also [21], Ch.7). In order to give a convenient description of the isotropy subgroups of these actions, we follow the presentation of [21] and introduce a subgroup Urs (H) = U(H) ∩ GLsr (H) consisting of all unitary operators from GLsr . For s = 2 this subgroup is denoted by Ur . Now the description of isotropy groups is available by the same way of reasoning which was applied in [21] for s = 2. Theorem 2.3. The subgroup Urs (H) acts transitively on Grrs (H) and the isotropy subgroup of the subspace H+ is isomorphic to U(H+ ) × U(H− ). From the existence of a polar decomposition for a bounded operator on H it follows that the subgroup Urs (H) is a retract of GLsr and it is straightforward to obtain similar conclusions for the actions of GLsr . Corollary 2.4. The group GLsr acts transitively on the Grassmannian Grrs (H) and the isotropy groups of this action are contractible.

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Thus such an action obviously defines a fibration with contractible fibers and it is well known that for such fibrations the total space (GLsr ) and the base (Grrs ) are homotopy equivalent [11]. Corollary 2.5. For any s ≥ 1, the Grassmannian Grrs and the group GLsr have the same homotopy type. In particular, GLr is homotopy equivalent to Grr . Actually, it can be shown that all groups GLsr have the same homotopy type for s ≥ 1. Thus all the above groups and Grassmannians have the same homotopy type. Notice that the homotopy groups of GLr and Grr over the field of complex numbers were computed in [17, 9, 21]. Thus we arrive at the folowing theorem which in particular gives an answer to a question posed in [3]. Theorem 2.6. For any s ∈ [1, ∞], the homotopy groups of the group GLsr and Grassmannian Grrs are given by the formulae: π0 ∼ = Z; π2k+1 ∼ = Z, π2k+2 = 0, k ≥ 0.

(5)

This is an important result which can be applied to the homotopy classification of linear conjugation problems [18, 7], but those applications are beyond the scope of the present paper. Therefore we switch to another general paradigm which emerged in the theory of restricted Grassmannians and Grassmannian embeddings of loop groups. As was shown in [13, 17], the topological study of loop groups can be performed in the framework of the theory of Fredholm structures [11]. Since the Grassmannian embeddings of loop groups establish a close relation between geometric properties of the loop groups and those of restricted Grassmannians, it became highly plausible that one should be able to construct geometrically meaningful Fredholm structures on restricted Grassmannians. Indeed, it was proved in [18] that loop groups can be endowed with natural Fredholm structures arising from the generalized linear conjugation problems. The main construction in [18] was given in the language of linear conjugation problems with coefficients in loop groups, which required a lot of preliminary considerations. Recently, we found another construction of Fredholm structure on the restricted Grassmannian which is more explicit and direct. As was explained in [19] and [7], Fredholm structures on the restricted Grassmannian induce the ones on loop groups and vice versa. Therefore the results presented in the last section provide simultaneously an alternative way of introducing Fredholm structures on loop groups which seems simpler than the one used in [13, 17].

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Fredholm structures on restricted Grassmannians

The aim of this section is to show that the restricted Grassmannians can be endowed with natural Fredholm structures. Before passing to precise formulations we recall necessary concepts from functional analysis. For a Banach space E, let L(E) denote the algebra of bounded linear operators in E endowed with the norm topology. Let F (E)(Fk (E)) denote the subset of Fredholm operators (of index k). Let also GL(E) stand for the group of units of L(E) and denote by GLc (E) the so-called Fredholm group of E defined as the set of all invertible operators from L(E) having the form ”identity plus compact”. Recall that a Fredholm structure on a smooth manifold M modeled on a (infinite dimensional) Banach space E is defined as a reduction of the structural group GL(E) of the tangent bundle T M to a subgroup GLc (E) [11]. In the sequel we only deal with the case when E = H is a separable Hilbert space but much of the following discussion is valid in a more general context. As GL(H) is contractible, F0 (H) is the classifying space for GLc (H) bundles [11]. For a Hilbert manifold M, defining a Fredholm structure on M is equivalent to constructing an index zero Fredholm map M → H [12]. It was also shown in [12] that a Fredholm structure on M can be constructed from a smooth map Φ : M → F0 (H), i.e., from a smooth family of index zero Fredholm operators parameterized by the points of M. This is actually the most effective way of constructing Fredholm structures which has already been used in [13, 17]. We are now going to present an explicit construction of such families on restricted Grassmannians. For simplicity we describe it for the restricted Grassmannian Grr which is an analytic Hilbert manifold modeled on the Hilbert space K2 (H) [21]. According to Theorem 1 the group Ur transitively acts on Grr and the isotropy subgroup of H+ in Ur is isomorphic to U(H+ ) × U(H− ). As was shown in [12] one can obtain Fredholm structure on Grr by constructing a family of index zero Fredholm operators in a Hilbert space parameterized by the points of Grr . We will construct a family of zero index Fredholm operators in H+ parameterized by the points of Grr . For V ∈ Grr , we first construct an element T ∈ Ur such that T (H+ ) = V . Let v : H+ → H be an isometry with image V and w : H− → H be an isometry with image W = V ⊥ . Then the mapping A = u ⊕ w : H+ ⊕ H− → H+ ⊕ H− is a unitary transformation in H such that A(H+ ) = V . Let us write it in the block form (Aij ) corresponding to the fixed decomposition H = H+ ⊕ H− . Then the left upper element T = A11 is a linear operator in H+ . As V ∈ Grr , from the definition of Grr it follows that T is a Fredholm operator in H+ , in other words, T ∈ F (H+ ). Consider now a component Gr0 of Grr consisting of subspaces L ∈ Grr such that ind P+ |L = 0. From the description of the connected components of the set of Fredholm pairs F p∗ given in Section 1 it follows that Gr0 is a connected component of Grr and all other connected components Grn are homeomorphic to Gr0 . Moreover, the group Ur

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permutes those components so that any geometric structure on one of the components can be transplanted to all of them. Thus it is sufficient to construct a Fredholm structure on Gr0 . Notice that the operator T constructed above is not unique so we cannot a priori assign it to V in a well-defined way. However from the description of the isotropy subgroups of Grr in Ur it follows that the totality of all such operators T has the structure of smooth fibration over Grr with contractible fiber. Thus by general results of infinite dimensional topology it has a global section S which can be chosen to be smooth [11]. This means that the assignment V 7→ S(V ) defines a smooth family of index zero Fredholm operators in H+ . Referring now to the aforementioned result from [12] we see that the family S(V ), V ∈ Gr0 , defines a Fredholm structure on Gr0 . In virtue of the said above this structure can be transplanted on all other connected components. Thus we have established the desired result. Theorem 3.1. The restricted Grassmannian of a polarized Hilbert space has a natural structure of an analytic Fredholm manifold modeled on the ideal of Hilbert-Schmidt operators K2 (H). By using a proper modification of the above construction one can show that a similar statement holds for each Grassmannian Grrs with s ≥ 1. Moreover, using the construction of an orientation bundle on the space of Fredholm pairs given in [1] one can show that, in the real case, the above Fredholm structure is orientable in the sense of [12]. This enables one to study the geometry and topology of restricted Grassmannians and Kato Grassmannians using methods of the theory of Fredholm structures [11, 12]. For the reason of space we confine ourselves to a few short remarks in this spirit. As was proved in [12], each Fredholm structure on manifold M induces a zero index Fredholm map of M into its model. It is natural to conjecture that such a map of Grr into K2 (H) can be obtained from our construction. It would be instructive to find an explicit description of that map. It would be also interesting to define the same Fredholm structure by an explicitly given atlas on Grr . Using the general methods of Fredholm structures theory, one can derive a number of immediate consequences of Theorem 3. For example, one can define Chern classes, fundamental classes of submanifolds, and so on in the spirit of [11, 13, 18]. Over the field of reals one can consider the maps between restricted Grassmannians arising from Fredholm operators in the ambient space and try to obtain their topological invariants using the degree theory for Fredholm manifolds developed in [12]. It is impossible to deal with any of the mentioned topics in a short paper like this one and so in conclusion we just express an intent to continue research in this direction.

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References [1] P. Abbondandolo and P. Majer: “Morse homology on Hilbert spaces“, Comm. Pure Applied Math., Vol. 54, (2001), pp. 689–760. [2] J. Avron, R. Seiler and B. Simon: “The index of a pair of projections”, J. Func. Anal., Vol. 120, (1994), pp. 220–237. [3] B. Bojarski: “Abstract linear conjugation problems and Fredholm pairs of subspaces“ (Russian), In: Differential and Integral equations. Boundary value problems. Collection of papers dedicated to the memory of Academician I, Vekua, Tbilisi University Press, Tbilisi, 1979, pp. 45–60. [4] B. Bojarski: “Some analytical and geometrical aspects of the Riemann-Hilbert transmission problem”, In: Complex analysis. Methods, trends, applications, Akad. Verlag, Berlin, 1983, pp. 97–110. [5] B. Bojarski: “The geometry of the Riemann-Hilbert problem”, Contemp. Math., Vol. 242, (1999), pp. 25–33. [6] B. Bojarski: “The geometry of Riemann-Hilbert problem II”, In: Boundary value problems and integral equations, World Scientific, Singapore, 2000, pp. 41–48. [7] B. Bojarski and G. Khimshiashvili: “Global geometric aspects of Riemann-Hilbert problems”, Georgian Math. J., Vol. 8, (2001), pp. 799–812. [8] B. Bojarski and A. Weber: “Generalized Riemann-Hilbert transmission and boundary value problems. Fredholm pairs and bordisms”, Bull. Polish Acad. Sci., Vol. 50, (2002), pp. 479–496. [9] B. Booss and K. Wojciechowsky: Elliptic boundary value problems for Dirac operators, Birkh¨auser, Boston, 1993. [10] M. Dupr´e and J. Glazebrook: “The Stiefel bundle of a Banach algebra”, Integr. Eq. Oper. Theory, Vol. 41, (2000), pp. 264–287. [11] J. Eells: “Fredholm structures”, Proc. Symp. Pure Math., Vol. 18, (1970), pp. 62–85. [12] J. Elworthy and A.Tromba: “Differential structures and Fredholm maps on Banach manifolds”, Proc. Symp. Pure Math., Vol. 15, (1970), pp. 45–94. [13] D. Freed: “The geometry of loop goups”, J. Diff. Geom., Vol. 28, (1988), pp. 223–276. [14] D. Freed: “An index theorem for families of Fredholm operators parametrized by a group”, Topology, Vol. 27, (1988), pp. 279–300. [15] T. Kato: Perturbation theory for linear operators, Springer, Berlin, 1980. [16] G. Khimshiashvili: On the topology of invertible linear singular integral operators, Springer Lecture Notes Math., Vol. 1214, (1986), pp. 211–230. [17] G. Khimshiashvili: On Fredholmian aspects of linear conjugation problems, Springer Lect. Notes Math., Vol. 1520, (1992), pp. 193–216. [18] G. Khimshiashvili: “Homotopy classes of elliptic transmision problems over C ∗ algebras”, Georgian Math. J., Vol. 5, (1998), pp. 453–468. [19] G. Khimshiashvili: “Geometric aspects of Riemann-Hilbert problems”, Mem. Diff. Eq. Math. Phys., Vol. 27, (2002), pp. 1–114. [20] G. Khimshiashvili: “Global geometric aspects of linear conjugation problems”, J. Math. Sci., Vol. 118, (2003), pp. 5400–5466.

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[21] A. Pressley and G. Segal: Loop groups, Clarendon Press, Oxford, 1986. [22] K. Wojciechowski: “Spectral flow and the general linear conjugation problem”, Simon Stevin Univ. J., Vol. 59, (1985), pp. 59–91.

CEJM 3(4) 2005 718–765

Covariance algebra of a partial dynamical system Bartosz Kosma Kwa´sniewski∗ Institute of Mathematics, University in Bialystok, ul. Akademicka 2, PL-15-424 Bialystok, Poland

Received 30 October 2004; accepted 11 August 2005 Abstract: A pair (X, α) is a partial dynamical system if X is a compact topological space and α : ∆ → X is a continuous mapping such that ∆ is open. Additionally we assume here that ∆ is closed and α(∆) is open. Such systems arise naturally while dealing with commutative C ∗ -dynamical systems. In this paper we construct and investigate a universal C ∗ -algebra C ∗ (X, α) which agrees with the partial crossed product [10] in the case α is injective, and with the crossed product by a monomorphism [22] in the case α is onto. The main method here is to use the description of maximal ideal space of a coefficient algebra, e α cf. [16, 18], in order to construct a larger system (X, e) where α e is a partial homeomorphism. Hence one may apply in extenso the partial crossed product theory [10, 13]. In particular, one generalizes the notions of topological freeness and invariance of a set, which are being used afterwards to obtain the Isomorphism Theorem and the complete description of ideals of C ∗ (X, α). c Central European Science Journals. All rights reserved.

Keywords: Crossed product, C ∗ -dynamical system, covariant representation, topological freeness MSC (2000): 47L65, 46L45, 37B99

Introduction In quantum theory the term covariance algebra (crossed product) means an algebra generated by an algebra of observables and by operators which determine the time evolution of a quantum system (a C ∗ -dynamical system), thereby the covariance algebra is an object which carries all the information about the quantum system, see [6, 18] (and the sources cited there) for this and other connections with mathematical physics. In pure mathematics C ∗ -algebras associated to C ∗ -dynamical systems proved to be useful in different ∗

E-mail: [email protected]

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fields: classification of operator algebras [25, 6, 23]; K-theory for C ∗ -algebras [4, 10, 20]; functional and functional differential equations [2, 3]; or even in number theory [17]. This multiplicity of applications and the complexity of the matter attracted many authors and caused an abundence of various approaches [25, 10, 20, 26, 1, 22, 12, 14]. In the present article we propose another approach which on one hand may seem to embrace a very special case but on the other hand: 1) unlike the other authors investigating crossed products (see discussions below) we do not require here any kind (substitute) of reversibility of the given system, 2) we obtain a rather thorough description of the associated covariance algebra, and also strong tools to study it, 3) we find points of contact of different approaches and thereby clarify the relations between them. In order to give the motivation of the construction of the crossed product developed in the paper we would like to present a simple example. Example. Consider the Hilbert space H = L2µ (R) where µ is the Lebesgue measure. Let A ⊂ L(H) be the C ∗ -algebra of operators of multiplication by continuous bounded functions on R that are constant on R− = {x : x ≤ 0}. Set the unitary operator U ∈ L(H) by the formula (Uf )(x) = f (x − 1), f ( · ) ∈ H. Routine verification shows that the mapping A ∋ a 7→ UaU ∗ is an endomorphism of A of the form UaU ∗ (x) = a(x − 1),

a( · ) ∈ A,

(1)

U ∗ aU(x) = a(x + 1),

a( · ) ∈ A.

(2)

and Clearly the mapping A ∋ a 7→ U ∗ aU does not preserve A. Let C ∗ (A, U) be the C ∗ -algebra generated by A and U. It is easy to show that C ∗ (A, U) = C ∗ (B, U) where B ⊂ L(H) is the C ∗ -algebra of operators of multiplication by continuous bounded functions on R that have limits at −∞. In addition we have that UBU ∗ ⊂ B

and

U ∗ BU ⊂ B

and the corresponding actions δ( · ) = U( · )U ∗ and δ∗ ( · ) = U ∗ ( · )U on B are given by formulae (1) and (2).

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Moreover C ∗ (A, U) = C ∗ (B, U) ∼ = B ×δ Z. Where in the right hand side stands the standard crossed product of B by the automorphism δ. This example shows a natural situation when the crossed product B ×δ Z is ’invisible’ at the begining (on the initial algebra A, δ acts as an endomorphism and δ∗ even does not preserve A) but after implementing a natural extension of A up to B, δ becomes an automorphism and leads to the crossed product structure. The aim of the paper is to investigate the general constructions of this type. We will also find out that the arising constructions are rather natural and one can come across them ’almost anywhere’, in particular they present the passage from the irreversible topological Markov chains to the reverible ones (see Propostition 2.8) and the maximal ideal spaces of the algebras of B type possess the solenoid structure (see Examples 2.12, 6.15). We deal here with C ∗ -dynamical systems where dynamics is implemented by a single endomorphism, hence a C ∗ -dynamical system is identified with a pair (A, δ) where A is a unital C ∗ -algebra and δ : A → A is a ∗ -endomorphism. Additionally we assume that A is commutative. Hence, in fact, we deal with topological dynamical systems. Indeed, using the Gelfand transform one can identify A with the algebra C(X) of continuous functions on the maximal ideal space X of A, and within this identification the endomorphism δ generates (see, for example, [16]) a continuous partial mapping α : ∆ → X where ∆ ⊂ X is closed and open (briefly clopen), and the following formula holds    a(α(x)) , x ∈ ∆ δ(a)(x) = , a ∈ C(X). (3)  0 , x∈ /∆

Therefore we have one-to-one correspondence between the commutative unital C ∗ -dynamical systems (A, δ) and pairs (X, α), where X is compact and α is a partial continuous mapping in which the domain is clopen. We shall call (X, α) a partial dynamical system. Usually covariance algebra is another name for the crossed product which in turn is defined in various ways [25, 23, 10, 22], though it seems more appropriate to define such objects as C ∗ -algebras with a universal property with respect to covariant representations [26, 1]. In the literature, cf. [25, 10, 20, 1], a covariant representation of (A, δ) is meant to be a triple (π, U, H) where H is a Hilbert space, π : A → L(H) is a representation of A by bounded operators on H, and U ∈ L(H) is such that π(δ(a)) = Uπ(a)U ∗ ,

for all a ∈ A,

plus eventually some other conditions imposed on U and π. If π is faithful we shall call (π, U, H) a covariant faithful representation. The covariant representations of a C ∗ dynamical system give rise to a category Cov(A, δ) where objects are the C ∗ -algebras C ∗ (π(A), U), generated by π(A) and U, while morphisms are the usual ∗ -morphisms

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φ : C ∗ (π(A), U) → C ∗ (π ′ (A), U ′ ) such that φ(π(a)) = π ′ (a), for a ∈ A,

and

φ(U) = U ′

(here (π, U, H) and (π ′ , U ′ , H ′) denote covariant representations of (A, δ)). In many cases the main interest is concentrated on the subcategory CovFaith(A, δ) of Cov(A, δ) for which objects are algebras C ∗ (π(A), U) where now π is faithful. The fundamental problem then is to describe a universal object in Cov(A, δ), or in CovFaith(A, δ), in terms of the C ∗ -dynamical system (A, δ). If such an object exists then it is unique up to isomorphism, and it shall be called a covariance algebra. It is well known [25] that, in the case that δ is an automorphism, the classic crossed product A ⋊δ Z is the covariance algebra of the C ∗ -dynamical system (A, δ). Being motivated by the paper [8], in which J. Cuntz discussed a concept of the crossed product by an endomorphism which is not an automorphism, many authors proposed theories of generalized crossed products with some kind of universality (see [24, 10, 20, 22, 12]). For instance, G. Murphy in [22] has proved that a corner of the crossed product of a certain direct limit is a covariance algebra of a system (A, δ) where δ is a monomorphism (in fact he has proved far more general result, see [22, Theorem 2.3]). R. Exel in [10] introduced a partial crossed product which can be applied also in the case δ is not injective, though generating a partial automorphism (see also [13, 20]). Nevertheless, in general the inter-relationship between the C ∗ -dynamical system and its covariance algebra is still not totally-established. In the approach developed in this paper we explore the leading concept of the coefficient algebra, introduced in [18]. The elements of this algebra play the role of Fourier’s coefficients in the covariance algebra, hence the name. The authors of [18] studied the C ∗ algebra C ∗ (A, U) generated by a ∗ -algebra A ⊂ L(H) and a partial isometry U ∈ L(H). They have defined A (in a slightly different yet equivalent form) to be a coefficient algebra of C ∗ (A, U) whenever A possess the following three properties U ∗ U ∈ A′ ,

UAU ∗ ⊂ A,

U ∗ AU ⊂ A,

(4)

where A′ denotes the commutant of A. Let us indicate that this concept appears, in more or less explicit form, in all the aforesaid articles: If U is unitary then (4) holds iff δ(·) = U(·)U ∗ is an automorphism of A, and thus in this case A can be regarded as a coefficient algebra of the crossed product A ⋊δ Z. For example in the paper [8], the UHF algebra Fn is a coefficient algebra of the Cuntz algebra On . Also the algebra A considered by Paschke in [24] is a coefficient algebra of the C ∗ -algebra C ∗ (A, S) generated by A and the isometry S. The algebra C α˜ defined in [22] as the fixed point algebra for dual action can be thought of as a generalized coefficient algebra of the crossed product C ∗ (A, M, α) of A by the semigroup M of injective endomorphisms. Thanks to [16], the main tool we are given is the description of maximal ideal spaces of certain coefficient algebras. More precisely, for any partial isometry U and unital commutative C ∗ -algebra A such that U ∗ U ∈ A′ and UAU ∗ ⊂ A we infer that (A, δ) is a C ∗ -dynamical system, where δ(·) = U(·)U ∗ . However A does not need to fulfill the

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third property from (4). The solution then is to pass to a bigger C ∗ -algebra B generated by {A, U ∗ AU, U 2∗ AU 2 , ...}. Then (B, δ) is a C ∗ -dynamical system and B is a coefficient algebra of C ∗ (B, U) = C ∗ (A, U), see [18]. In this case the authors of [16] managed to ’estimate’ the maximal ideal space M(B) of B in terms of (A, δ), or better to say, in terms of the generated partial dynamical system (X, α). Fortunately, the full description of M(B) is obtained [16, 3.4] by a slight strengthening of assumptions - namely by assuming that the projection U ∗ U belongs not only to commutant A′ but to A itself. The partial dynamical system (M(B), α e), corresponding to (B, δ), is thus completely determined by (X, α). Two important facts are to be noticed: α e is a partial homeomorphism, and ∗ U U ∈ A implies that the image α(∆) of the partial mapping α is open, see Section 1 for details. In Section 2, to an arbitrary partial dynamical system (X, α) such that α(∆) is open e α we associate another partial dynamical system (X, e) such that: 1) α e is a partial homeomorphism,

e → X such that the equality Φ ◦ α 2) there exist a continuous surjection Φ : X e = α◦Φ holds wherever it makes sense (see diagram (18)), e α 3) if α is injective then Φ becomes a homeomorphism, that is (X, e) ∼ = (X, α).

e α This authorizes us to call (X, e) the reversible extension of (X, α). In the case α is e e α onto, X is a projective limit (see Proposition 2.10) and thus (X, e) generalizes the known construction. In Section 3 we find out that all the objects of CovFaith(A, δ) have the same (up to e We denote this C ∗ isomorphism) coefficient C ∗ -algebra whose maximal ideal space is X. algebra by B. Then we construct a coefficient ∗ -algebra B0 (the closure of B0 is B) with the help of which we express the interrelations between the covariant representations of e where δe is an endomorphism associated to the partial homeomorphism (A, δ) and (B, δ) α e. In particular we show that, if δ is injective, or equivalently α is onto, then we have a natural one-to-one correspondence between aforementioned representations. In general this correspondence is maintained only if we constrain ourselves to covariant faithful representations. In Section 4 we define C ∗ (A, δ) = C ∗ (X, α) to be the partial crossed product of e by a partial automorphism generated by the partial homeomorphism α B = C(X) e. We ∗ show that C (A, δ) is the universal object in CovFaith(A, δ), and in the case that δ is injective, it is also universal when considered as an object of Cov(A, δ). Therefore we call it a covariance algebra. Section 5 is devoted to two important notions in C ∗ -dynamical systems theory: topological freeness and invariant sets. Classically, these notions were related only to homeomorphisms, but recently they have been adopted (generalized), by authors of [13], to work with partial homeomorphisms, see also [19]. Inspired by this line of development we present here the definitions of topological freeness and invariance under a partial mapping which include also noninjective partial mappings. Let us mention that, for instance, in [15] appears also the definition of topologically free irreversible dynamical system, but

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the authors of [15] attach to dynamical systems different C ∗ -algebras than we do, hence they are in need of a different definition. We show that there exists a natural bijection e and that the between closed α-invariant subsets of X and closed α e-invariant subsets of X partial dynamical system (X, α) is topologically free if and only if its reversible extension e α (X, e) is topologically free. Section 6 contains two important results. Namely, we establish a one-to-one correspondence between the ideals in C ∗ (X, α) and closed invariant subsets of X generalizing Theorem 3.5 from [13]. Then we present a version of the Isomorphism Theorem, cf. [2, Theorem 7.1], [3, Chapter 2], [13, Theorem 2.6], [18, Theorem 2.13], which says that all objects of CovFaith(A, δ) are isomorphic to C ∗ (A, δ) whenever the corresponding system (X, α) is topologically free.

1

Preliminaries. Maximal ideal space of a coefficient C ∗-algebra

We start this section by fixing some notation. Afterwards, we present and discuss briefly the results of [16] in order to present our methods and motivations. We finish this section with Theorem 1.8 to be used extensively in the sequel. Throughout this article A denotes a commutative unital C ∗ -algebra, X denotes its maximal ideal space (i.e. a compact topological space), δ is an endomorphism of A, while α stands for a continuous partial mapping α : ∆ → X where ∆ ⊂ X is clopen and the formula (3) holds. We adhere to the convention that N = 0, 1, 2, ..., and when dealing with partial mappings we follow the notation of [16], i.e.: for n > 0, we denote the domain of αn by ∆n = α−n (X) and its image by ∆−n = αn (∆n ); for n = 0, we set ∆0 = X and thus, for n, m ∈ N, we have αn : ∆n → ∆−n , (5) αn (αm (x)) = αn+m(x),

x ∈ ∆n+m .

(6)

We recall that in terms of the multiplicative functionals of A, α is given by x ∈ ∆1 ⇐⇒ x(δ(1)) = 1,

(7)

α(x) = x ◦ δ,

(8)

x ∈ ∆1 .

For the purpose of the present section we fix (only in this section) a faithful representation of A, i.e. we assume that A is a C ∗ -subalgebra of L(H) where L(H) is an algebra of bounded linear operators on a Hilbert space H. Additionally we assume that endomorphism δ is given by the formula δ(a) = UaU ∗ ,

a ∈ A,

for some U ∈ L(H) and so U is a partial isometry (note that there exists a correspondence between properties of U and the partial mapping α, cf. [16, 2.4]). In that case, as it makes sense, we will consider δ also as a mapping on L(H). There is a point in studying together with δ(·) = U(·)U ∗ one more mapping δ∗ (b) = U ∗ bU,

b ∈ L(H),

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which in general maps a ∈ A onto an element outside the algebra A and hence, even if we assume that U ∗ U ∈ A′ , we need to pass to a bigger algebra in order to obtain an algebra satisfying (4) . Proposition 1.1. [18, Proposition 4.1] If δ(·) = U(·)U ∗ is an endomorphism of A, S S∞ ∗n n ∗ ∗n n U ∗ U ∈ A′ and B = C ∗ ( ∞ n=0 U AU ) is a C -algebra generated by n=0 U AU , then B is commutative and both the mappings δ : B → B and δ∗ : B → B are endomorphisms. The elements of the algebra B play the role of coefficients in a C ∗ -algebra C ∗ (A, U) generated by A and U, [18, 2.3]. Hence the authors of [18] call B a coefficient algebra. It is of primary importance that B is commutative and that we have a description of its maximal ideal space, denoted here by M(B), in terms of the maximal ideals in A, see [16]. Let us recall it. With every x e ∈ M(B) we associate a sequence of functionals ξxen : A → C , n ∈ N, defined by the condition ξxen (a) = δ∗n (a)(e x),

a ∈ A.

(9)

S n The sequence ξxen determines x e uniquely because B = C ∗ ( ∞ n=0 δ∗ (A)). Since δ∗ is an endomorphism of B the functionals ξxen are linear and multiplicative on A. So either ξxen ∈ X (X is the spectrum of A) or ξxen ≡ 0. It follows then that the mapping M(B) ∋ x e → (ξxe0, ξxe1 , ...) ∈

∞ Y

(X ∪ {0})

(10)

n=0

is an injection and the following statement is true, see Theorems 3.1 and 3.3 in [16]. Theorem 1.2. Let δ(·) = U(·)U ∗ be an endomorphism of A, U ∗ U ∈ A′ , and α : ∆1 → X be the partial mapping determined by δ. Then the mapping (10) defines a topological S b embedding of M(B) into topological space ∞ N =0 XN ∪ X∞ . Under this embedding we have ∞ ∞ [ [ b N ∪ X∞ XN ∪ X∞ ⊂ M(B) ⊂ X N =0

N =0

where

bN = {e X x = (x0 , x1 , ..., xN , 0, ...) : xn ∈ ∆n , α(xn ) = xn−1 , 1 ≤ n ≤ N}, X∞ = {e x = (x0 , x1 , ...) : xn ∈ ∆n , α(xn ) = xn−1 , 1 ≤ n}.

bN : xN ∈ XN = {e x = (x0 , x1 , ..., xN , 0, ...) ∈ X / ∆−1 }, S b The topology on ∞ N =0 XN ∪ X∞ is defined by a fundamental system of neighborhoods bN given by of points x e∈X bN : |ai (xN ) − ai (yN )| < ε, i = 1, ..., k} O(a1 , ..., ak , ε) = {e y∈X

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and respectively of x e ∈ X∞ by

O(a1 , ..., ak , n, ε) = {e y∈

∞ [

N =n

where ε > 0, ai ∈ A and k, n ∈ N.

725

bN ∪ X∞ : |ai (xn ) − ai (yn )| < ε, i = 1, ..., k} X

Remark 1.3. The topology on X is weak∗ . One immediately sees then (see (10)), that the S bN ∪X∞ is in fact the product topology inherited from Q∞ (X ∪{0}) topology on N ∈N X n=0 where {0} is clopen. The foregoing theorem gives us an estimate of M(B) and aiming at sharpening that result we need to strengthen the assumptions. If we replace the condition U ∗ U ∈ A′ with the stronger one U ∗ U ∈ A, (11) then the full information on B is carried by the pair (A, δ), cf. [16, Theorem 3.4]. Theorem 1.4. Under the assumptions of Theorem 1.2 with U ∗ U ∈ A′ replaced by U ∗ U ∈ A we get ∞ [ M(B) = XN ∪ X∞ . N =0

This motivates us to take a closer look at condition (11). Firstly, let us observe [16, 3.5] that if U ∗ U ∈ A′ then δ is an endomorphism of the C ∗ S ∗n n algebra A1 = C ∗ (A, U ∗ U) and since C ∗ ( ∞ n=0 U A1 U ) = B one can apply the preceding theorem to the algebra A1 for the full description of M(B). This procedure turns out to be very fruitful in many situations. Example 1.5. Let the elements of A be the operators of multiplication by periodic sequences of period n, on the Hilbert space l2 (N), and let U be the co-isometry given by [Ux](k) = x(k + 1), for x ∈ l2 (N), k ∈ N. Then X = {x0 , ..., xn−1 } and α(xk ) = xk+1 (mod n) . If for k = 0, ..., n − 1 we write (∞, k) = (xk , xk−1 , ..., x1 , x0 , xn−1 , xn−2 , . . . ) and (N, k) = (xk , xk−1 , ..., x1 , x0 , xn−1 , ..., xn−r , 0, 0, . . . ) | {z } N

where N − r ≡ k (mod n) (for each N there are n pairs (N, k)), then from Theorem 1.2 we get {∞} × {0, 1, ..., n − 1} ⊂ M(B) ⊂ N × {0, 1, ..., n − 1} where N = N ∪ {∞} is a compactification of the discrete space N. In order to describe M(B) precisely let us pass to the algebra A1 = C ∗ (A, U ∗ U). As U ∗ U is the operator of multiplication by (0, 1, 1, ...) the elements of A1 are the operators of multiplication by

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sequences of the form (a, h(0), h(1), ...) where a is arbitrary and h(k + n) = h(k), for all k ∈ N. In an obvious manner (with a slight abuse of notation) we infer the spectrum of A1 to be {y, x0 , ..., xn−1 }, and the mapping generated by δ considered as an endomorphism of A1 acts as follows: α(xk ) = xk+1 (mod n) and α(y) = x0 . Abusing notation once again and putting (N, k) = (xk , xk−1 , ..., x1 , x0 , xn−1 , ..., x0 , y, 0, 0, ...) | {z } N

where N ≡ k (mod n) (for each N there is now the only one pair (N, k)), in view of Theorem 1.4 we have
M(B) = {(N, k) ∈ N × {0, 1, ..., n − 1} : N ≡ k (mod n)} ∪ {∞} × {0, 1, ..., n − 1} ,

so M(B) can be imagined as a spiral subset of the cylinder N×{0, 1, ..., n−1}, (see Figure 1).

q

(0, 0)

q

(1, 1)

.

(2, 2)

q

.

q

.

q

(n, 0)

q

(n+1, 1)

(n+2, 2)

q (∞, n−1) q  (∞, 0)PPP P q (∞, 1)

.. .. . q

(∞, 2)

Fig. 1 Maximal ideal space of the coefficient algebra from Example 1.5.

Condition (11) is closely related to the openness of ∆−1 (as ∆1 is compact and α is continuous ∆−1 is always closed). Proposition 1.6. Let P∆−1 be the projection corresponding to the characteristic function χ∆−1 . If U ∗ U ∈ A then ∆−1 is open and U ∗ U = P∆−1 . If U ∗ U ∈ A′ , ∆−1 is open and A acts nondegenerately on H, then U ∗ U 6 P∆−1 . Proof. Let U ∗ U ∈ A. We show that the image ∆−1 of the set ∆1 of functionals satisfying (7) under the mapping (8) is the set of functionals x ∈ X satisfying x(U ∗ U) = 1. Let x′ ∈ ∆1 , that is x′ (UU ∗ ) = 1. Putting x = α(x′ ) we have x(U ∗ U) = x′ (δ(U ∗ U)) = x′ (UU ∗ UU ∗ ) = x′ (UU ∗ )x′ (UU ∗ ) = 1. Now, let x ∈ X be such that x(U ∗ U) = 1. We define on δ(A) a multiplicative functional x′ (b) := x(U ∗ bU), b ∈ δ(A). For b = δ(a), a ∈ A, we then have x′ (δ(a)) = x(U ∗ UaU ∗ U) = x(U ∗ U)x(a)x(U ∗ U) = x(a).

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x′ is therefore well defined. Since x′ (δ(1)) = x(1) = 1 it is non zero and there exists its extension x′ ∈ X on A (see [9, 2.10.2]). It is clear that x′ ∈ ∆1 and α(x′ ) = x. Hence x ∈ ∆−1 . Thus we have proved that x ∈ ∆−1 ⇐⇒ x(U ∗ U) = 1, which means that U ∗ U ∈ A is the characteristic function of ∆−1 . It follows then that ∆−1 is clopen. Now, let ∆−1 be open. Then χ∆−1 ∈ C(X) and δ(χ∆−1 ) = χα−1 (∆−1 ) = χ∆1 . As A acts nondegenerately, rewriting this equation in terms of operators we have UP∆−1 U ∗ = UU ∗ .

(12)

Letting Hi = U ∗ UH be the initial and Hf = UU ∗ H be the final space of U we get U ∗ : Hf → Hi is an isomorphism and U : Hi → Hf is its inverse. Taking arbitrary h ∈ Hf and applying the both sides of (12) to it we obtain UP∆−1 U ∗ h = h, and hence P∆−1 Hi = Hi , that is U ∗ U 6 P∆−1 .  Note. The inequality in the second part of the preceding proposition can not be replaced by equality. In order to see that consider for instance A and U from Example 1.5. By virtue of Proposition 1.1 the mappings δ and δ∗ are endomorphisms of the C ∗ -algebra B. With the help of the presented theorems we can now find the form of the partial mappings they generate. We shall rely on the fact [16, 2.5] expressed by the coming proposition. Proposition 1.7. Let δ(·) = U(·)U ∗ and δ∗ (·) = U ∗ (·)U be endomorphisms of A and let α be the partial mapping of X generated by δ. Then ∆1 and ∆−1 are clopen and α : ∆1 → ∆−1 is a homeomorphism. Moreover, the endomorphism δ∗ is given on C(X) by the formula    a(α−1 (x)) , x ∈ ∆−1 δ∗ (a)(x) = (13)  0 , x∈ / ∆−1 Finally we arrive at the closing theorem.

Theorem 1.8. Let the hypotheses of Theorem 1.2 hold. Then i) the sets e 1 = {(x0 , ...) ∈ M(B) : x0 ∈ ∆1 }, ∆

e −1 = {(x0 , x1 , ...) ∈ M(B) : x1 6= 0} ∆

are clopen subsets of M(B), e1 → ∆ e −1 ii) the endomorphism δ generates on M(B) the partial homeomorphism α e:∆ given by the formula α e(x0 , ...) = (α(x0 ), x0 , ...),

e 1, (x0 , ...) ∈ ∆

(14)

e −1 → ∆ e1 iii) the partial mapping generated by δ∗ is the inverse of α e, that is α e−1 : ∆ where e −1 . α e−1 (x0 , x1 ...) = (x1 , ...), (x0 , x1 ...) ∈ ∆ (15)

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Proof. We rewrite Proposition 1.7 in terms of Theorem 1.2. Let x e = (x0 , x1 , ...) ∈ M(B). From (7) we get e 1 ⇐⇒ x x e∈∆ e(UU ∗ ) = 1,

e −1 ⇐⇒ x x e∈∆ e(U ∗ U) = 1.

However, the definition (9) of functionals ξxen = xn implies that x e(UU ∗ ) = 1 ⇐⇒ x0 (UU ∗ ) = 1, and x e(U ∗ U) = 1 ⇐⇒ x1 (1) = 1, which proves i). The mapping α e generated by δ on M(B) (see (8)), is given by the composition: α e(e x) ≡ n n e x e ◦ δ. So, let x e = (x0 , x1 , ...) ∈ ∆1 , then the sequence of functionals ξxe satisfies: ξxe (a) = a(xn ), a ∈ A, n ∈ N. Now let us consider an analogous sequence of functionals ξαne (ex) defining the point α e(e x) = (x0 , x1 , ...). For n > 0 we have a(xn ) = ξαne (ex) (a) = α e(e x)(δ∗n (a)) = x e(δ(δ∗n (a))) = x e(UU ∗n aU n U ∗ ) = =x e(UU ∗ )e x(δ∗n−1 (a))e x(U ∗ U) = x e(δ∗n−1 (a)) = ξxen−1 (a) = a(xn−1 ),

while for n = 0 we have

a(x0 ) = ξα0e (ex) (a) = α e(e x)(a) = x e(δ(a)) = ξxe0 (δ(a)) = δ(a)(x0 ) = a(α(x0 )).

Thus we infer that α e(e x) = (α(x0 ), x0 , ...). By Proposition 1.7, α e−1 is the inverse to mapping α e. Hence we get (15). 

2

Reversible extension of a partial dynamical system

One of the most important consequences of Theorems 1.4 and 1.8 is that although the algebra B is relatively bigger than A and its structure depends on U and U ∗ (U and U ∗ need not be in A) the C ∗ -dynamical system (B, δ) still can be reconstructed by means of the intrinsic features of (A, δ) itself (provided (11) holds). Therefore in this section, we make an effort to investigate effectively this reconstruction and, as it is purely topological, we forget for the time being about its algebraic aspects. e α Once having the system (X, α), we will construct a pair (X, e): e - a counterpart of the maximal ideal space obtained in Theorem • a compact space X 1.4 (or a ’lower estimate’ of it, see Theorem 1.2), and • a partial injective mapping α e - a counterpart of the mapping from Theorem 1.8. e α We then show some useful results about the structure of (X, e). In particular, we cale α e and culate (X, e) for topological Markov chains, and show the interrelation between X projective limits. The most interesting case occurs when the injective mapping α e has an open image. We shall call such mappings partial homeomorphisms, cf [13]. More precisely, a partial homeomorphism is a partial mapping which is injective and has open image. Let us recall that by a partial mapping of X we always mean a continuous mapping α : ∆1 → X such that ∆1 ⊂ X is clopen, and so if α is a partial homeomorphism, then

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α−1 : ∆−1 → X is a partial mapping of X in our sense, that is ∆−1 is clopen. e actually becomes As we shall see if ∆−1 is open then α e is a partial homeomorphism and X the spectrum of a certain coefficient algebra (see Theorem 3.3). That is the reason why we shall often assume the openness of ∆−1 . However in this section we do not make it a standing assumption in order to get to know better the role of it and the condition (11), cf. Proposition 1.6.

e α 2.1 The system (X, e)

Let us fix a partial dynamical system (X, α) and let us consider a disjoint union X ∪ {0} of the set X and the singleton {0} (we treat here 0 as a symbol rather than the number). We define {0} to be clopen and hence X ∪ {0} is a compact topological space. We will e to be a subset define X ∞ Y e X⊂ (X ∪ {0}) n=0

e represent of the product of ℵ0 copies of the space X ∪ {0} where the elements of X anti-orbits of the partial mapping α. Namely we set

where

e= X

∞ [

N =0

XN ∪ X∞

(16)

XN = {e x = (x0 , x1 , ..., xN , 0, ...) : xn ∈ ∆n , xN ∈ / ∆−1 , α(xn ) = xn−1 , n = 1, ..., N}, X∞ = {e x = (x0 , x1 , ...) : xn ∈ ∆n , α(xn ) = xn−1 , n > 1}. Q e is the one induced from the space ∞ (X ∪ {0}) equipped The natural topology on X n=0 e ⊂ Q∞ (∆n ∪ {0}), the topology on with the product topology, cf. Remark 1.3. Since X n=0 e can also be regarded as the topology inherited from Q∞ (∆n ∪ {0}). X n=0 e the extension of X under α, or Definition 2.1. We shall call the topological space X briefly the α-extension of X.

e Moreover, Theorem 2.2. The subset X∞ is compact and the subsets XN are clopen in X. the following conditions are equivalent: a) ∆−1 is open. e is compact. b) X c) X0 is compact. d) XN is compact for every N ∈ N. Q Proof. As the sets ∆n , n ∈ N, are clopen, by Tichonov’s theorem, the space ∞ n=0 (∆n ∪ {0}) is compact, and to prove the compactness of X∞ it suffices to show that X∞ is a Q closed subset of ∞ n=0 (∆n ∪ {0}), or equivalently that its complement is open. To this

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Q end, let x e = (x0 , x1 , ...) ∈ ∞ and suppose that x e∈ / X∞ . We show that x e n=0 (∆n ∪ {0}) Q∞ e has an open neighborhood contained in n=0 (∆n ∪ {0}) \ X∞ . In view of the definition of X∞ we have two possibilities: 1) there is n > 0 such that xn = 0, 2) there is n > 0 such that xn ∈ ∆n , and α(xn ) 6= xn−1 , Q If 1) holds then the set Ve = ∞ k=0 Vk where Vk = ∆k ∪ {0}, for k 6= n, and Vn = {0}, is an open neighborhood of x e and Ve ∩ X∞ = ∅. Let us suppose now that 2) holds. We may also suppose that xn−1 6= 0 (xn−1 ∈ ∆n−1 ). Hence there exist two disjoint open subsets V1 , V2 ⊂ ∆n−1 such that α(xn ) ∈ V1 and Q xn−1 ∈ V2 . Clearly, the set Ve = ∞ k=0 Vk where Vk = ∆k ∪ {0}, for k 6= n − 1, n, and −1 Vn−1 = V2 , Vn = α (V1 ), is an open neighborhood of x e, and V1 ∩ V2 = ∅ guarantees that e V ∩ X∞ = ∅. Fix N ∈ N. To prove that XN is open we recall that ∆n , n ∈ N, are clopen and ∆−1 is closed. Hence ∆n \ ∆−1 , n ∈ N, are open, and it is easy to see that e ∩ (∆0 × ∆1 × ... × ∆N −1 × ∆N \ ∆−1 × (∆N +1 ∪ {0}) × ...). XN = X

e It is also closed because its complement is the sum of Hence XN is an open subset of X. e : xN = 0} and Ve2 = {(x0 , x1 ...) ∈ X e : xN +1 6= 0}. two open sets: Ve1 = {(x0 , x1 ...) ∈ X We prove now the equivalence of a), b), c) and d). e in an analogous a)⇒b). Suppose that ∆−1 is open. We prove the compactness of X Q∞ e fashion as we proved the compactness of X∞ . Let x e = (x0 , x1 , ...) ∈ n=0 (∆n ∪ {0}) \ X. e we have the three possibilities: In view of the definition of X 1) there are n, m ∈ N such that xn = 0 and xn+m ∈ ∆n+m (xn+m 6= 0), 2) there is n > 0 such that xn ∈ ∆n , and α(xn ) 6= xn−1 , 3) for some n > 0 we have xn ∈ ∆n ∩ ∆−1 , and xn+1 = 0. Q Let us suppose that 1) holds. Then the set Ve = ∞ k=0 Vk where Vk = ∆k ∪ {0}, for k 6= n, n + m, and Vn = {0}, Vn+m = ∆n+m , is an open neighborhood of x e. It is clear e e that none of the points from X belongs to V . The same argumentation as the one concerning X∞ shows that in the case 2) x e lies in Q∞ e the interior of n=0 (∆n ∪ {0}) \ X. Q If we suppose that 3) holds, then the set Ve = ∞ k=0 Vk where Vk = ∆k ∪ {0}, for k 6= n, n + 1, and Vn = ∆n ∩ ∆−1 , Vn+1 = {0}, is an open neighborhood of x e (here we use the e e openness of ∆−1 ). Clearly V does not contain any point from X. b)⇒c). X0 is closed and hence it is compact. c)⇒a). Suppose that ∆−1 is not open. Then X \ ∆−1 is not closed and hence it is not compact. Thus there is an open cover {Vi }i∈I of X \ ∆−1 which does not admit a finite e : x0 ∈ Vi }, for i ∈ I, we get an open cover of X0 subcover. Defining Vei = {(x0 , x1 ...) ∈ X which does not admit a finite subcover. Hence X0 is not compact. Thus we see that a)⇔b)⇔c). As b)⇒d) and d)⇒c) are obvious, the proof is complete. e depends on α. For instance, if α is surjective then Xn , n ∈ N, It is interesting how X e = X∞ , in this case X e can be defined as a projective limit, see Proposition are empty and X

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e onto X given by the 2.10. If α is injective then a natural continuous projection Φ of X formula Φ(x0 , x1 , ...) = x0 (17) becomes a bijection and, as we will see, in the case that ∆−1 is open even a homeomore is from X. phism. But the farther from injectivity α is, the farther X

e →X Proposition 2.3. Let (X, α) be such a system for which α is injective. Then Φ : X is a homeomorphism if and only if ∆−1 is open.

e is not compact and hence not homeProof. If ∆−1 is not open then by Theorem 2.2, X omorphic to X. Suppose then that ∆−1 is open. Then α : ∆1 → X is an open mapping because α is a homeomorphism of compact set ∆1 onto the compact set ∆−1 . We only need to show that the mapping Φ−1 is continuous or, which is the same, that Φ is open. e in a appropriate way. To see this it is enough to look at a subbase for the topology in X e⊂X e be of the form Indeed, let U e=X e ∩ (U0 × U1 × ... × UN × (∆N +1 ∪ {0}) × (∆N +2 ∪ {0}) × ...) U

where Ui ⊂ X ∪ {0}, i = 1, ..., N, are open. Without loss of generality we can assume that Ui ⊂ ∆i ∪ {0}, i = 1, ..., N. There are the two possibilities: 1) If 0 ∈ / UN then the set UN −1 ∩ α(UN ) is open and   e=X e ∩ (U0 × U1 × ... × UN −1 ∩ α(UN ) × (∆N ∪ {0}) × (∆N +1 ∪ {0}) × ...) U

2) If 0 ∈ UN then the set UN −1 ∩ α(UN \ {0}) ∪ UN −1 \ ∆−1 ∪ {0} is open and   e e U = X ∩ (U0 × U1 × ... × UN −1 ∩ α(UN \ {0}) ∪ UN −1 \ ∆−1 ∪ {0} × (∆N ∪ {0}) × ...).

Q e=X e ∩ ∞ Vk where Vk = Applying the above procedure N times we conclude that U k=0 e ) = V0 . ∆k ∪ {0}, for k > 0, and V0 ⊂ X ∪ {0} is a certain open set. It is obvious that Φ(U Thus Φ is an open mapping and the proof is complete. 

Example 2.4. Consider the dynamical system (X, α) where X = [0, 1] and α(x) = q · x, for fixed 0 < q < 1 and any x ∈ [0, 1]. In this situation α is injective and ∆−1 = [0, q] is e with an interval [0, 1] but the topology on not open. As Φ is bijective, we can identify X e differs from the natural topology on [0, 1]. It is not hard to check that the topology X e = [0, 1] is generated by intervals [0, a), (a, b), (b, 1], where 0 < a < b < 1, and on X e is not compact and Φ is not a homeomorphism. singletons {q k }, k > 0. Thus X e associated with α. It Now, we would like to investigate a partial mapping α e on X seems very natural to look for a partial mapping α e such that Φ ◦ α e = α ◦ Φ wherever the e 1 := Φ−1 (∆1 ) and superposition α ◦ Φ makes sense. If such α e exists then its domain is ∆ e −1 := Φ−1 (∆−1 ). Moreover, as Φ is continuous, we get its image is contained in ∆ e 1 = Φ−1 (∆1 ) = {e e : x0 ∈ ∆1 } ∆ x = (x0 , x1 , ...) ∈ X

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is clopen while e −1 = Φ−1 (∆−1 ) = {e e : x0 ∈ ∆−1 } ∆ x = (x0 , x1 , ...) ∈ X

e −1 is open too. It is not a surprise that there always is closed and if ∆−1 is open then ∆ exists a homeomorphism α e such that the following diagram α e e 1 −→ e −1 ∆ ∆

Φ↓

(18)

↓Φ

α

∆1 −→ ∆−1 commutes. However, the commutativity of the diagram (18) does not determine the homeomorphism α e uniquely.

e1 → ∆ e −1 given by the formula Proposition 2.5. The mapping α e:∆ α e(e x) = (α(x0 ), x0 , ...),

e1 x e = (x0 , ...) ∈ ∆

(19)

e such is a homeomorphism (and hence if ∆−1 is open α e is a partial homeomorphism of X) that the diagram (18) is commutative. Proof. The inverse of α is given by the formula α e−1 ((x0 , x1 , ...)) = (x1 , x2 , ...). The e 1 ∩ (U0 × U1 × ...)) = X e ∩ (X × U0 × ...) and α e −1 ∩ straightforward equations α e (∆ e−1 (∆ −1 e ∩ ((α (U0 ) ∩ U1 ) × U2 × ...), and the definition of the topology (U0 × U1 × U2 × ...)) = X e imply that α in X e and α e−1 are continuous and hence they are homeomorphisms. The commutativity of the diagram (18) is obvious.  In this manner we can attach to every pair (X, α) such that ∆−1 is open another system e e and if α is a partial homeomorphism (X, α e) where α e is a partial homeomorphism of X e α then systems (X, α) and (X, e) are topologically equivalent via Φ, see Proposition 2.3. In particular case of a classical irreversible dynamical system, that is when α is a covering e α mapping of X, α e is a full homeomorphism and hence (X, e) is a classical reversible dynamical system. This motivates us to coin the following definition. Definition 2.6. Let (X, α) be a partial dynamical system and let ∆−1 be open. We say e α e and α that the pair (X, e), where X e are given by (16) and (19) respectively, is a reversible extension of (X, α). Example 2.7. It may happen that two different partial dynamical systems have the same reversible extension. Let (X, α) and (X ′ , α′) be given by the relations: X = {x0 , x1 , x2 , y2 }, ∆1 = X \ {x0 }, α(y2 ) = α(x2 ) = x1 , α(x1 ) = x0 ; X ′ = {x′0 , x′1 , x′2 , y1′ , y2′ }, ∆′1 = X ′ \ {x′0 } and α′ (x′2 ) = x′1 , α′ (y2′ ) = y1′ , α′ (y1′ ) = α′ (x′1 ) = x′0 ; or by the diagrams: q

x0

q  x2   ) qi PP x1P P q

y2

q  x′   ) 1 qi PP x′0P P q

y1′

q

x′2 q

y2′

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e consist of the following points x Then X e0 = (x0 , x1 , x2 , 0, ...), x e1 = (x1 , x2 , 0, ...), x e2 = e′ (x2 , 0, ...), ye0 = (x0 , x1 , y2 , 0, ...), ye1 = (x1 , y2 , 0, ...) and ye2 = (y2 , 0, ...). Similarly X is the set of points x e′0 = (x′0 , x′1 , x′2 , 0, ...), x e′1 = (x′1 , x′2 , 0, ...), x e′2 = (x′2 , 0, ...), ye0′ = e α (x′0 , y1′ , y2′ , 0, ...), ye1′ = (y1′ , y2′ , 0, ...) and ye2′ = (y2′ , 0, ...). Hence the systems (X, e) and ′ ′ e (X , α e ) are given by the same diagram x eq 0 x eq 1 x eq2 x eq ′0 x eq ′1 x eq′2 yeq 0

yeq 1

yeq2

yeq 0′

yeq 1′

yeq2′

2.2 Topological Markov chains, projective limits and hyperbolic attractors It is not hard to give an example of a partial dynamical system which is not a reversible extension of any ’smaller’ dynamical system, though its dynamics is implemented by a partial homeomorphism. Yet many reversible dynamical systems arise from irreversible ones as their reversible extensions. In order to see that we recall first the topological Markov chains and then the hyperbolic attractors. Let A = (A(i, j))i,j∈{1,...,N } be a square matrix with entries in {0, 1}, and such that no row of A is identically zero. We associate with A two dynamical systems (XA , σA ) and (X A , σA ). The one-sided Markov subshift σA acts on the compact space XA = {(xk )k∈N ∈ {1, ..., N}N : A(xk , xk+1 ) = 1, k ∈ N} (the topology on XA is the one inherited from the Cantor space {1, ..., N}N ) by the rule (σA x)k = xk+1 ,

for k ∈ N, and x ∈ XA .

Unless A is a permutation matrix σA is not injective, and σA is onto if and only if every column of A is non-zero. The two-sided Markov subshift σ A acts on the compact space X A = {(xk )k∈Z ∈ {1, ..., N}Z : A(xk , xk+1 ) = 1, k ∈ Z} and is defined by (σ A x)k = xk+1 ,

for k ∈ Z, and x ∈ X A .

Mapping σ A is what is called a topological Markov chain and abstractly can be defined as an expansive homeomorphism of a completely disconnected compactum. If we assume that not only the rows of A but also the columns are not identically zero then σA is onto and we have eA , σ Proposition 2.8. Let A have no zero columns and let (X eA ) be the reversible extension of (XA , σA ). Then eA , σ (X eA ) ∼ = (X A , σ A ).

eA . Then x Proof. Let x e ∈ X e = (x0 , x1 , ...) where xn = (xn,k )k∈N ∈ XA is such that m σA (xn+m ) = xn , n, m ∈ N. Thus, xn+m,k+m = xn,k , for k, n, m ∈ N, or in other words m1 − n1 = m2 − n2 =⇒ xn1 ,m1 = xn2 ,m2 .

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We see that the sequence x ∈ σ A such that xk = xn,m , m − n = k ∈ Z, carries the full information about x e. Hence, defining Υ by the formula Υ(e x) = (..., xn,0 , ..., x1,0 , x˙ 0,0 , x0,1 , ..., x0,n , ...)

(20)

eA → X A . where dot over x0,0 denotes the zero entry, we get an injective mapping Υ : X It is evident that Υ is surjective and can be readily checked that it is also continuous, whence Υ is a homeomorphism. Finally let us recall that σ eA (e x) = (σA (x0 ), x0 , ...) and σA (x0 ) = (x0,1 , x0,2 , ...) and thus Υ(e σA (e x)) = (..., xn,0 , ..., x1,0 , x0,0 , x˙ 0,1 , ..., x0,n , ...) = σ A (Υ(e x))

eA , σ which says that (X eA ) and (σ A , X A ) are topologically conjugate by Υ.



The shift σA has a clopen image of the form

σA (XA ) = {(xk )k∈N ∈ XA :

N X

A(x, x0 ) > 0}.

x=1

Thus, if A has at least one zero column then σA is not onto, and since σ A is always onto, S eA , σ eA = systems (X eA ) and (X A , σA ) cannot be conjugate. In fact, X n∈N XA,n ∪ XA,∞ , see Definition 2.1, and in the same manner as in the proof of Proposition 2.8 we may define a homeomorphism Υ from the subset XA,∞ = {(x0 , x1 , ...) : xn ∈ XA , σA (xn ) = xn−1 , n > 1} onto X A such that Υ(e σA (e x)) = σ A (Υ(e x)) for x e ∈ XA,∞ , that is (XA,∞ , σ eA ) ∼ = (X A , σ A ).

eA we need to add a In order to build a homeomorphic image of the whole space X countable number of components to X A . We may do it by embedding X A into another space X A′ associated with the larger alphabet {0, 1, ..., N} and a larger matrix A′ = (A′ (i, j))i,j∈{0,1,...,N }. Proposition 2.9. Let A′ = (A′ (i, j))i,j∈{0,1,...,N } be given by    A(i, j), if i, j ∈ {1, ..., N}, A′ (i, j) = 1, if i = 0 and either j = 0 or j-th column of A is zero ,   0, otherwise, +

+

and let XA+′ = (xk )k∈Z ∈ XA+′ : x0 6= 0. Then σ A′ (X A′ ) ⊂ X A′ and + eA , σ (X eA ) ∼ = (X A′ , σ A′ ).

eA = Proof. Let us treat X A as a subset of {0, 1, ..., N}Z and recall that X XA,∞ , and we have the homeomorphism Υ : XA,∞ → X A , cf. (20). We put Υ(e x) = (..., 0, 0, xn−1,0, ..., x1,0 , x˙ 0,0 , x0,1 , ..., x0,n , ...)

S

n∈N

XA,n ∪

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735

for x e = (x0 , x1 , ..., xn−1 , 0, ...) ∈ XA,n where xm = (xm,k )k∈N ∈ XA , m = 0, ..., n − 1. In the same fashion as in the proof of Proposition 2.8 one checks that the mapping eA → {0, 1, ..., N}Z is injective and the equality Υ(e Υ : X σA (e x)) = σ A′ (Υ(e x)) holds for e every x e ∈ XA . Moreover, XA,n is mapped by Υ onto the set Z

{(xk )k∈Z ∈ {0, 1, ..., N} : xk = 0, k < −n;

N X i=1

A(i, xn ) = 0; A(xk , xk+1 ) = 1, k > −n}

S + eA , σ denoted by X A,n . Thus (X eA ) ∼ = ( n∈N X A,n ∪ X A , σ A′ ). It is clear that X A′ = S  n∈N X A,n ∪ X A and hence the proof is complete.

The proof of Proposition 2.8 is actually the proof of the probably known fact that if σA σA is onto, then X A is the projective (inverse) limit of the projective sequence XA ←− σA XA ←− ... . Let us pick out the relationship between α-extensions and projective limits. For that purpose (and also for future needs) we introduce some terminology. e n the domain of α As α e is a partial homeomorphism, we denote by ∆ en , n ∈ Z. For n ∈ N we have e n = {e e : x0 ∈ ∆n } = Φ−1 (∆n ), ∆ x = (x0 , x1 , ...) ∈ X e −n = {e e : xn 6= 0} ⊂ Φ−1 (∆−n ). ∆ x = (x0 , x1 , ...) ∈ X

e −n −→ With the help of Φ and α e−1 , we define the family of projections Φn = Φ ◦ α e−n : ∆ ∆n , for n ∈ N. We have

e −n . Φn (e x) = (Φ ◦ α e−n )(x0 , x1 , ..., xn , ...) = xn , x e∈∆ T e Since X∞ = ∞ n=1 ∆−n , the mappings Φn are well defined on X∞ , and the following statements are straightforward. Proposition 2.10. The system (X∞ , Φn )n∈N is the projective limit of the projective sequence (∆n , αn )n∈N where αn = α|∆n : X∞ = ← lim −−(∆n , αn ). e = lim (∆n , αn ). Corollary 2.11. If α is onto, then X ←−−

The above statement provides us with many interesting examples of reversible extensions because projective (inverse) limit spaces commonly appear as attractors in dynamical systems (this was observed for the first time by R. F. Willams, see [27]). We recall here the classic example. Example 2.12 (Solenoid). Let S 1 = {z ∈ C : |z| = 1} be the unit circle in the complex plane and let α be the expanding endomorphism of S 1 given by α(z) = z 2 ,

z ∈ S 1.

1 Then the projective limit ← lim solenoid, that is an −−(S , α) is homeomorphic to 1Smale’s 2 attractor of the mapping F acting on the solid torus T = S × D , where D 2 = {z ∈ C : |z| ≤ 1}, by F (z1 , z2 ) = (z12 , λz2 + 21 z1 ) where 0 < λ < 12 is fixed.

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Fig. 2 The image of the solid torus under F is a solid torus which wraps twice around itself.

T Namely the solenoid is the set S = k∈N F k (T ), see Fig. 2, and the reversible extension of (S 1 , α) is equivalent to (S, F |S ), see e.g. [7].

e 2.3 Decomposition of sets in X

Before the end of this section we introduce a certain idea which enables us to ’decompose’ e ⊂X e into the family {Un }n∈N of subsets of X. We shall need this device in a subset U Section 5. e⊂X e be a subset of α-extension of X and let n ∈ N. We shall Definition 2.13. Let U call the set e ∩∆ e −n ) Un = Φn (U

e an n-section of U.

 e then U e is a subset of U0 × It is evident that if {Un }n∈N is the family of sections of U  e but in general the opposite relation does not hold (U1 ∪ {0}) × ... × (Un ∪ {0}) × ... ∩ X (see Example 2.16). Fortunately we have the following true statements. T Proposition 2.14. If α is injective on the inverse image of ∆−∞ := n∈N ∆−n , that is e⊂X e we have for every point x ∈ ∆−∞ we have |α−1 (x)| = 1, then for every subset U   e = U0 × (U1 ∪ {0}) × ... × (Un ∪ {0}) × ... ∩ X e U (21) e , n ∈ N. where Un is the n-section of U

e We show that Proof. Let x e = (x0 , x1 , ...) ∈ (U0 × (U1 ∪ {0}) × ... × (Un ∪ {0}) × ...) ∩ X. T e . Indeed, if x e −n , then x x e∈U e∈ / X∞ = n∈N ∆ e = (x0 , ..., xN , 0, ...) where xN ∈ ∆N \ ∆−1 , e ∩∆ e −N ), U e must contain and thus x e is uniquely determined by xN . As xN ∈ UN = ΦN (U x e. If x e ∈ X∞ then xn ∈ ∆−∞ for all n ∈ N, and x e is uniquely determined by x0 ∈ U0 . Thus e. x e∈U 

B.K. Kwa´sniewski / Central European Journal of Mathematics 3(4) 2005 718–765

737

Theorem 2.15. Let (X, α) be a partial dynamical system such that ∆−1 is open. Then e is uniquely determined by its sections {Vn }n∈N via formula every closed subset Ve ⊂ X (21). e be closed, that is compact (see Theorem 2.2), and let x Proof. Let Ve ⊂ X e = (x0 , x1 , ...) ∈ e (V0 × (V1 ∪ {0}) × ... × (Vn ∪ {0}) × ...) ∩ X. We show that x e ∈ Ve . If x e ∈ / X∞ then (see the argument in the proof of Proposition 2.14) we immediately get x e ∈ Ve . Thus we only need to consider the case when x e ∈ X∞ . For that purpose we define e e e e n }n∈N is the decreasing Dn = {e y = (y0, y1 , ...) ∈ X : yn = xn } ∩ V , n ∈ N. Clearly {D family of closed nonempty subsets of the compact set Ve . Hence \ e n = {e D x} ∈ Ve n∈N

and the proof is complete.



Example 2.16. For the sake of illustration of the preceding statements and to see that they cannot be sharpen let us consider a dynamical system given by the diagram xq 1 -x0q 



or equivalently by relations X = ∆1  = {x0 , x1 }, α(x1 ) = α(x0 ) = x0 (or equivalently 1 0 let α = σA and X = XA where A =  ). Then ∆−∞ = {x0 } and |α−1 (x0 )| = 2, 10 e consists of elements x therefore Proposition 2.14 can not be used. The space X en = e (x0 , ..., x0 , x1 , 0, ...), n ∈ N, and x e∞ = (x0 , ..., x0 , ...). Hence it is convenient to identify X | {z } n

with the compactification N = N ∪ {∞} of the discrete space N. Under this identification α e is given by α e(n) = n + 1, n ∈ N, α e(∞) = ∞.

It is clear that all the sections of the subset N ⊂ N are equal to X. As N is not closed in N, Theorem 2.15 does not work here. Indeed, we have   N = X × (X ∪ {0}) × ... × (X ∪ {0}) × ... 6= N.

3

Covariant representations and their coefficient algebra

The aim of this section is to study the interrelations between the covariant representations e α of C ∗ -dynamical systems corresponding to (X, α) and its reversible extension (X, e). Our main tool will be Theorem 1.4 and hence, cf. Proposition 1.6, from now on we shall always assume that the image ∆−1 of the partial mapping α is open.

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e possesses a certain universal property with First we show that the algebra B = C(X) respect to covariant faithful representations of (X, α). Afterwards, we construct a dense ∗-subalgebra of B with the help of which we investigate the structure of B and endomorphisms induced by α e and α e−1 . Finally we show that there is a one-to-one correspondence e α between the covariant faithful representations of (X, α) and (X, e), and in the case α is onto this correspondence is true for all covariant (not necessarily faithful) representations.

3.1 Definition and basic result Let us recall that A = C(X) and δ is combined with α by (3). We denote by CK (X) the algebra of continuous functions on X vanishing outside a set K ⊂ X. We start with the definition of covariant representation, cf. [25, 10, 20, 1]. Definition 3.1. A covariant representation of a C ∗ -dynamical system (A, δ), or of the partial dynamical system (X, α), is a triple (π, U, H) where π : A → L(H) is a representation of A on Hilbert space H and U ∈ L(H) is a partial isometry whose initial space is π(C∆−1 (X))H and whose final space is π(C∆1 (X))H. In addition it is required that Uπ(a)U ∗ = π(δ(a)),

for a ∈ A.

If the representation π is faithful we call the triple (π, U, H) a covariant faithful representation. Let CovRep(A, δ) denote the set of all covariant representations and CovFaithRep(A, δ) the set of all covariant faithful representations of (A, δ). Remark 3.2. As ∆1 and ∆−1 are clopen, the projections P∆1 and P∆−1 corresponding to the characteristic functions χ∆1 and χ∆−1 belong to A. Thus for every covariant representation (π, U, H) we see that UU ∗ = π(P∆1 ) and U ∗ U = π(P∆−1 ) belong to π(A), cf. (11). We shall see in Corollary 3.13 that every C ∗ -dynamical system (A, δ) possesses a covariant faithful representation, and hence the sets CovRep (A, δ) and CovFaithRep (A, δ) are nonempty. Now we reformulate the main result of [16] in terms of covariant representations. The point is that for every covariant representation (π, U, H) of (A, δ) the condition (11) holds, whence if π is faithful then by Theorem 1.4 the maximal ideal space of the C ∗ -algebra S ∗n n e C ∗( ∞ n=0 U π(A)U ) is homeomorphic to α-extension X of X.

e be the α-extension of X. Theorem 3.3. Let (π, U, H) ∈ CovFaithRep (A, δ) and let X Then ∞ [
∗ e C U ∗n π(A)U n ∼ = C(X). n=0

∗

In other words, the coefficient C -algebra of C ∗ (π(A), U) is isomorphic to the algebra of continuous functions on α-extension of X, cf. [18]. Moreover this isomorphism maps an operator of the form π(a0 ) + U ∗ π(a1 )U + ... + U ∗N π(aN )U N , where a0 , a1 , ..., aN ∈ A, onto

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739

e such that a function b ∈ C(X)

b(e x) = a0 (x0 ) + a1 (x1 ) + ... + aN (xN ),

e and we set an (xn ) = 0 whenever xn = 0. where x e = (x0 , ...) ∈ X

S ∗n n Proof. By Theorem 1.4 the maximal ideal space of C ∗ ( ∞ n=0 U π(A)U ) is homeomorS e Hence C ∗ ( ∞ U ∗n π(A)U n ) ∼ e Taking into account formulas (9),(10) phic to X. = C(X). n=0 we obtain the postulated form of this isomorphism. Indeed, if x e = (x0 , x1 , ...) is a charS∞ ∗ ∗n n acter on C ( n=0 U π(A)U ) then x e(

N X n=0

∗n

n

U π(an )U ) =

N X n=0

∗n

n

x e(U π(an )U ) =

N X n=0

ξxen (an )

=

where ξxen (a) = x e(U ∗n π(a)U n ), cf. (9), and xn = 0 whenever ξxen ≡ 0.

N X

an (xn )

n=0



e can be regarded as the universal (in fact From the above it follows that B = C(X) unique) coefficient C ∗ -algebra for covariant faithful representations. In case δ is injective (that is α is onto), the universality of B is much ’stronger’, see Proposition 3.7.

3.2 Coefficient C ∗-algebra e a coefficient C ∗ -algebra We shall present now a certain dense ∗ -subalgebra of B = C(X), which frequently might be more convenient to work with. The plan is to construct an algebra B0 ⊂ l1 (N, A) and then take the quotient of it by certain ideal. The result will be naturally isomorphic to a ∗ -subalgebra of B. First, let us observe that if we set An := δ n (1)A, n ∈ N, then we obtain a decreasing family {An }n∈N , of closed two-sided ideals in A. Since the operator δ n (1) corresponds to the characteristic function χ∆n ∈ C(X), one can consider An as C∆n (X). Let B0 denote the set consisting of sequences a = {an }n∈N where an ∈ An , n ∈ N, and only a finite number of functions an is non zero. Namely B0 = {a ∈

∞ Y

n=0

An : ∃N >0 ∀n>N

an ≡ 0}.

Let a = {an }n>0 , b = {bn }n>0 ∈ B0 and λ ∈ C. We define the addition, multiplication by scalar, convolution multiplication and involution on B0 as follows (a + b)n = an + bn ,

(22)

(λa)n = λan ,

(23)

(a · b)n = an

n X

δ j (bn−j ) + bn

j=0

n X

δ j (an−j ),

(24)

j=1

∗

(a )n = an .

(25)

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These operations are well defined and seems very familiar, except maybe the multiplication of two elements from B0 . We point out here that the index in one of the sums of (24) starts running from 0. Proposition 3.4. The set B0 with operations (22), (23), (24), (25) becomes a commutative algebra with involution. Proof. It is clear that operations (22), (23) define the structure of vector space on B0 and that operation (25) is an involution. The rule (24) is less easy to show its properties. Commutativity and distributivity can be checked easily but in order to prove the associativity we must strain ourselves quite a lot. Let a, b, c ∈ B0 . Then ((a · b) · c)n = (a · b)n (a · (b · c))n = an where (a · b)n

n X

n X j=0

j

δ (cn−j ) = [an

j=0

n X

j

δ (cn−j ) + cn

j=0

j=1

= an

δ ((b · c)n−j ) + (b · c)n

n X

k

δ (bn−k ) + bn

cn

n X j=1

= cn

n X

δ (an−j )

j=1

k

n X

j

δ (an−j

j=1

j

k=j

j=0

(b · c)n

n X j=1

j

k

δ (an−k )]

n X

δ j (cn−j )

j=0

δ k (an−k )δ j (cn−j )

n−j X

k

δ (bn−j−k ) + bn−j

n X

j

δ ((b · c)n−j ) = an

δ (an−j ) = bn

n X

k

n X

k

δ (an−k )) = cn

δ k (an−j−k ))

n X

δ j (an−j )δ k (bn−k ).

k=1,j=1

δ k (bn−k )δ j (cn−j )

k,j=0

j

δ (an−k )δ (cn−j ) + cn

k=0,j=1

n−j X k=1

k=j+1

Simultaneously by analogous computation an

j=1

k=0

δ (bn−k ) + δ (bn−j )

n X

δ j (an−j ),

k=0,j=1

δ ((a · b)n−j ) = cn n X

n X

δ k (bn−k )δ j (cn−j ) + bn

j

j

n X

n X

k=1

k,j=0

and

δ j ((a · b)n−j ),

j

k=0

n X

n X

n X

δ j (an−j )δ k (bn−k ).

k=1,j=1

Thus, ((a · b) · c)n = (a · (b · c))n and the n-th entry of the sequence a · b · c is of the form an

n X

k,j=0

δ k (bn−k )δ j (cn−j ) + bn

n X

k=0,j=1

δ k (an−k )δ j (cn−j ) + cn

n X

k=1,j=1

δ k (an−k )δ j (bn−j ).

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 Let us define a morphism ϕ : B0 → B. To this end, let a = {an }n∈N ∈ B0 and e We set x e = (x0 , x1 , ...) ∈ X. ∞ X ϕ(a)(e x) = an (xn ), (26) n=0

where an (xn ) = 0 whenever xn = 0. The mapping ϕ is well defined as only a finite number of functions an , n ∈ N, is non zero.

Theorem 3.5. The mapping ϕ : B0 → B given by (26) is a morphism of algebras with involution and the image of ϕ is dense in B, that is ϕ(B0 ) = B. Proof. It is clear that ϕ is a linear mapping preserving an involution. We show that ϕ e and let N > 0 be such that for is multiplicative. Let a, b ∈ B0 and x e = (x0 , x1 , ...) ∈ X every m > N we have am = bm = 0. Using the fact that αj (xn ) = xn−j we obtain N N h n n i X X X X j j ϕ(a · b)(e x) = (a · b)n (xn ) = an δ (bn−j ) + bn δ (an−j ) (xn ) n=0

n=0

j=0

j=1

n n N X X X   bn−j (xn−j ) + bn (xn ) an−j (xn−j ) an (xn ) = n=0

=

N X

n=0,j=0

j=1

j=0

an (xn )bj (xj ) =

N X n=0

an (xn ) ·

N X j=0

  bj (xj ) = ϕ(a) · ϕ(b) (e x).

e we use the Stone-Weierstrass theorem. It is To prove that ϕ(B0 ) is dense in B = C(X) clear that ϕ(B0 ) is a self-adjoint subalgebra of B and as a = (1, 0, 0, ...) ∈ B0 , we get ϕ(a) = 1 ∈ B, that is ϕ(B0 ) contains the identity. Thus, what we only need to prove is e that B0 separates points of X. e Then there exists n ∈ Let x e = (x0 , x1 , ...) and ye = (y0 , y1 , ...) be two distinct points of X. N such that xn 6= yn and by Urysohn’s lemma there exists a function an ∈ C∆n (X ∪ {0}) such that an (xn ) = 1 and an (yn ) = 0. Taking a ∈ B0 of the form a = (0, ..., 0, an , 0, ...) | {z } n

we see that ϕ(a)(e x) = 1 and ϕ(a)(e y ) = 0. Thus the proof is complete.



Let us consider the quotient space B0 /Ker ϕ and the quotient mapping φ : B0 /Ker ϕ → B0 , that is φ(a + Ker ϕ) = ϕ(a). Clearly φ is an injective mapping onto a dense ∗ subalgebra of B. In what follows we make use of the following notation B0 := φ(B0 /Ker ϕ)

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[a] := φ(a + Ker ϕ),

a ∈ B0

Definition 3.6. We shall call B0 the coefficient ∗ -algebra of a dynamical system (A, δ). We will write [a] = [a0 , a1 , ...] ∈ B0 for a = (a0 , a1 , a2 , ...) ∈ B0 . The natural injection A ∋ a0 −→ [a0 , 0, 0, ...] ∈ B0 enables us to treat A as an C -subalgebra of B0 and hence also of B: ∗

A ⊂ B0 ⊂ B,

B0 = B.

e −n −→ ∆n (see subsection 2.2) one can embed into B0 not Using the mappings Φn : ∆ only A but all the subalgebras An = C∆n (X), n ∈ N. Indeed, if we define Φ∗n : An → B to act as follows    a ◦ Φn , x e −n e∈∆ Φ∗n (a) = [0, ..., 0, a, 0, ...] = , | {z }  e −n  0 , x e ∈ / ∆ n S then clearly Φ∗n are injective. Moreover we have C ∗ ( n∈N Φ∗n (An )) = B, and in the case δ is a monomorphism, that is α is surjective, {Φ∗n (An )}n∈N forms an increasing family of S algebras and B0 = n∈N Φ∗n (An ). We are exploiting this fact in the coming proposition. Proposition 3.7. If δ is injective then B is the direct limit lim −−→An of the sequence ∞ (An , δn )n=0 where δn = δ|An , n ∈ N.

∞ n Proof. Let B = − lim −→An be the direct limit of the sequence (An , δn )n=0 , and let ψ : An → B be the natural homomorphisms, see for instance [21]. It is straightforward to see that the diagram

An

δ

n −→

An+1

ց Φ∗n

↓ Φ∗n+1 B

commutes and hence there exists a unique homomorphism ψ : B → B such that the diagram An

ψn

−→ ց Φ∗n

B ↓ψ B

S commutes. It is evident that ψ is a surjection ( n∈N Φ∗n (An ) generates B) and as Φ∗n (An ) is increasing and, Φ and ψ are injective. Therefore ψ is an isomorphism and the proof is complete.  The preceding proposition points out the relationship between our approach and the approach presented (among the others) by G. J. Murphy in [22]. We shall discuss this

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743

relationship in the sequel, see Remark 4.10. e given by the formulae Let us now proceed and consider endomorphisms of B = C(X)      a(e  a(e e −1 e1 α−1 (e x)) , x e∈∆ α(e x)) , x e∈∆ e δ(a)(x) = δe∗ (a)(x) = (27)   e e 0  0 ,x e∈ / ∆−1 ,x e∈ / ∆1

e1 → ∆ e −1 is a canonical partial homeomorphism of X e defined by formula where α e : ∆ e 1 and ∆ e −1 belong to A ⊂ (19). What is important is that characteristic functions of ∆ B0 . Indeed, we have χ∆e 1 = [χ∆1 , 0, 0, ...], χ∆e 1 = [χ∆−1 , 0, 0, ...] ∈ A (see remark after e n of the mapping α Definition 3.6). Furthermore, the domain ∆ en , n ∈ Z, is clopen and it is just an easy exercise to check that for n ∈ N we have χ∆e n = [χ∆n , 0, 0, ...], χ∆e −n = [0, 0, ..., 0, ∆n , ...] ∈ B0 . In particular χ∆e −1 = [χ∆−1 , 0, 0, ...] = [0, χ∆1 , 0, ...]. We | {z } n

are now ready to give an ‘algebraic’ description of δe and δe∗ .

Proposition 3.8. Endomorphisms δe and δe∗ preserve ∗ -subalgebra B0 ⊂ B and for a = (a0 , a1 , a2 , ...) ∈ B0 we have e δ([a]) = [δ(a0 ) + a1 , a2 , a3 , ...],

δe∗ ([a]) = [0, a0 δ(1), a1 δ 2 (1), ...].

(28)

e In order to prove the first equality in (28) it is Proof. Let x e = (x0 , x1 , x2 , ...) ∈ X. e 1 we have enough to notice that for x e∈∆ δ([a])(e x) = [a](e α(e x)) = [a](α(x0 ), x0 , x1 , ...) = a0 (α(x0 )) + a1 (x0 ) + a2 (x1 ) + ...

= [δ(a0 ) + a1 , a2 , a3 , ...](x0 , x1 , x2 , ...) = [δ(a0 ) + a1 , a2 , a3 , ...](e x) e 1 both sides of the left hand side equation in (28) are equal to zero. and for x e∈ /∆ e −1 then In the same manner we show the validity of the remaining equality. If x e∈∆ δ∗ ([a])(e x) = [a](e α−1 (e x)) = [a](x1 , x2 , x3 ...) = a0 (x1 ) + a1 (x2 ) + a2 (x3 ) + ... = [0, a0 δ(1), a1 δ 2 (1), ...](x0 , x1 , x2 , ...) = [0, a0 δ(1), a1 δ 2 (1), ...](e x)

e 1 then both sides of the right hand side equation in (28) are equal to zero. and if x e∈ /∆

Example 3.9. The dynamical system (X, α) from Example 2.16 corresponds to the C ∗ e with N dynamical system (A, δ) where A = C({x0 , x1 }) and δ(a) ≡ a(x0 ). We identify X as we did before. Then, since a = [a0 , a1 , ..., aN , 0, ...], ak ∈ A, k = 0, ...N, is a continuous e = N we can regard it as a sequence which has a limit. One readily checks function on X Pn−1 that this sequence has the following form: a(n) = k=0 ak (x0 ) + an (x1 ) for n = 0, ...N, PN ∗ and a(n) = k=0 ak (x0 ) for n > N. Hence B0 is the -algebra of the eventually constant sequences, in particular A consist of sequences of the form (a, b, b, b, ...), a, b ∈ C. We have B0 = {(a(n))n∈N : ∃N ∈N ∀n,m>N a(n) = a(m)}, B = {(a(n))n∈N : ∃a(∞)∈C lim a(n) = a(∞)} n→∞

and within these identifications δe is the forward and δe∗ is the backward shift.

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3.3 The interrelations between covariant representations The construction of the ∗ -algebra B0 enables us to excavate the inverse of the isomorphism from Theorem 3.3, and what is more important it enables us to realize that every covariant e faithful representation of (A, δ) gives rise to a covariant faithful representation of (B, δ).

Theorem 3.10. Let (π, U, H) ∈ CovFaithRep (A, δ). Then there exists an extension
S ∗n n π of π onto the coefficient algebra B such that π : B → C ∗ ∞ is an n=0 U π(A)U isomorphism defined by [a0 , ..., aN , 0, ...] −→ π(a0 ) + U ∗ π(a1 )U + ... + U ∗N π(aN )U N . e that is Moreover, we have (π, U, H) ∈ CovFaithRep (B, δ), e π(δ(a)) = Uπ(a)U ∗ ,

π(δe∗ (a)) = U ∗ π(a)U,

a ∈ B.

(29)

Proof. In view of Theorem 3.3 it is immediate that π is an isomorphism. By Theorem 1.8 and by the form of endomorphisms δe and δe∗ , see (27), we get (29).  We can give a statement somewhat inverse to the above.

e and let π be the restriction of π onto Theorem 3.11. Let (π, U, H) ∈ CovRep (B, δ) e then A. Then (π, U, H) ∈ CovRep (A, δ). Moreover if (π, U, H) is in CovFaithRep (B, δ) (π, U, H) is in CovFaithRep (A, δ) and the extension of π mentioned in Theorem 3.10 coincides with π. Proof. Recall that for a ∈ A we write [a, 0, 0, ...] ∈ B and thus (see also Proposition 3.8) we get e π(δ(a)) = π([δ(a), 0, 0, ...]) = π(δ([a, 0, 0, ...])) = Uπ([a, 0, 0, ...])U ∗ = Uπ(a)U ∗ . U ∗ U = π(χ∆e 1 ) = π([χ∆1 , 0, 0, ...]) = π(χ∆1 ),

UU ∗ = π(χ∆e −1 ) = π([χ∆− 1 , 0, 0, ...]) = π(χ∆−1 ).

Hence (π, U, H) ∈ CovRep (A, δ). For [a0 , ..., aN , 0, ...] ∈ B0 we have, cf. Proposition 3.8,

π([a0 , ..., aN , 0, ...]) = π(a0 + δe∗ (a1 )+...+ δe∗N (aN )) = π(a0 )+U ∗ π(a1 )U +...+U ∗N π(aN )U N .

Hence the second part of the theorem follows.



Corollary 3.12. There is a natural bijection between CovFaithRep (A, δ) and e CovFaithRep (B, δ).

e onto C e (X), e whence we have The endomorphism δe maps isomorphically C∆e −1 (X) ∆1 a ∗-isomorphism between two closed two-sided ideals in B. In [10] R. Exel calls such

B.K. Kwa´sniewski / Central European Journal of Mathematics 3(4) 2005 718–765

745

mappings partial automorphisms (of B), see [10, Definition 3.1]. He also considers covariant representations of partial automorphisms and his definition of those objects agrees with Definition 3.1 in the case that α is a partial homeomorphism. Moreover, R. Exel proves in [10, Theorem 5.2] the existance of covariant faithful representation of a partial automorphism which automatically gives us Corollary 3.13. The set CovFaithRep (A, δ) is not empty. e is not empty by [10, Theorem 5.2]. Proof. The set CovFaithRep (B, δ)



The former of the preceding corollaries is not true for not faithful representations e is larger than CovFaithRep (A, δ), (see Example 3.16). In general the set CovRep (B, δ) and there appears a problem with prolongation of π from A to B when π is not faithful. Fortunately in view of Proposition 3.7 we have the following true statement. Theorem 3.14. If δ : A → A is a monomorphism then for any (π, U, H) ∈ CovRep (A, δ) e such that π([a0 , ..., aN , 0, ...]) = π(a0 ) + U ∗ π(a1 )U + there exist (π, U, H) ∈ CovRep (B, δ) ... + U ∗N π(aN )U N . Proof. Let (π, U, H) ∈ CovRep (A, δ). Let us notice that as δ is injective ∆−1 = X and hence U is an isometry (see Remark 3.2). Now, consider the C ∗ -algebra

S∞ S∞ ∗n n ∗ ∗n n C∗ U π(A)U and define the family of mappings π : A → C U π(A)U , n n n=0 n=0 n ∈ N, by the formula πn (a) = U ∗n π(a)U n ,

a ∈ An .

Then the following diagram An

δ

n −→

An+1

ց πn

↓ πn+1 C∗

S∞

n=0

U ∗n π(A)U n




commutates. Hence, according to Proposition 3.7 there exists a unique C ∗ -morphism π such that the diagram An

Φ

∗n −→

B ↓π

ց πn C∗ commutes. The hypotheses now follows.

S∞

n=0

U ∗n π(A)U n






Corollary 3.15. If δ is a monomorphism then the mapping π → π|A establishes a e onto CovRep (A, δ). bijection from CovRep (B, δ)

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e may In case δ is not a monomorphism two different covariant representations of (B, δ) induce the same covariant representation of (A, δ). Example 3.16. Not to look very far let us take the system (X, α) from Example 2.7. The corresponding C ∗ -dynamical system is (A, δ) where A = C({x0 , x1 , x2 , y2}) ∼ = C4 and δ(a) = (0, ax0 , ax1 , ax1 ) for a = (ax0 , ax1 , ax2 , ay2 ) ∈ A. The coefficient algebra is B = C({e x0 , x e1 , x e2 , ye0 , ye1 , ye2 }) ∼ = C6

and we check that, for a ∈ A,

[a, 0, ...] = (ax0 , ax1 , ax2 , ax0 , ax1 , ay2 ),

[0, aδ(1), 0, ...] = (0, ax0 , ax1 , 0, ax0 , ax1 ),

[0, 0, aδ 2(1), 0, ...] = (0, 0, ax0 , 0, 0, ax0 ) and [0, 0, ..., 0, aδ N (1), 0...] ≡ 0, for N > 2. e xe , axe , axe , aye , aye , aye ) = (0, axe , axe , 0, aye , aye ). It is now straightMoreover, we have δ(a 0 1 2 0 1 2 0 1 0 1

0 forward that (π 1 , U, C2) and (π 2 , U, C2 ) where U =  1

and

0 , 0









 axe0 0  π1 (axe0 , axe1 , axe2 , aye0 , aye1 , aye2 ) =   0 axe1  aye0 0  π 2 (axe0 , axe1 , axe2 , aye0 , aye1 , aye2 ) =   0 aye1

e which induce the same covariant representation are covariant representations of (B, δ)    ax0 0  (π, U, C2) of (A, δ) where π(ax0 , ax1 , ax2 , ay2 ) =  . 0 ax1

4

Covariance algebra

In this section we introduce the title object of the paper. We recall the definition of the partial crossed product, cf. [10, 20], and then define the covariance algebra of (A, δ) to e We give a number of examples of be the partial crossed product associated with (B, δ). such algebras, and finally show (justify the definition) that the covariant algebra is the universal object in the category of covariant faithful representations of (A, δ) and in the case δ is injective also in the category of covariant (not necessarily faithful) representations of (A, δ).

B.K. Kwa´sniewski / Central European Journal of Mathematics 3(4) 2005 718–765

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4.1 The algebra C ∗(X, α) Let us recall that a partial automorphism of a C ∗ -algebra C is a mapping θ : I → J where I and J are closed two-sided ideals in C and θ is a ∗-isomorphism, cf. [10]. If a partial automorphism θ is given then for each n ∈ Z we let Dn denote the domain of θ−n with the convention that D0 = C and θ0 is the identity automorphism of C. Letting L = {a ∈ l1 (Z, C) : a(n) ∈ Dn } and defining the convolution multiplication, involution, and norm as follows ∞   X
(a ∗ b)(n) = θk θ−k a(k) b(n − k) k=−∞

∗

(a )(n) = θn (a(−n)∗ ) ∞ X kak = ka(n)k n=−∞

∗

we equip L with a Banach -algebra structure. The universal enveloping C ∗ -algebra of L is called the partial crossed product (or the covariance algebra) for the partial automorphism θ and is denoted by C ⋊θ Z, see [10, 20]. e defines the partial automorphism It is clear that the partial homeomorphism α e of X e → C e (X) e of the coefficient C ∗ -algebra B = C(X) e (we shall not distinguish δe : C∆e −1 (X) ∆1 e ⊂ B). The between the endomorphism δe given by (27) and its restriction to C e (X) ∆−1

definition to follow anticipates Theorem 4.7.

Definition 4.1. The covariance algebra C ∗ (X, α) of a partial dynamical system (X, α) is the partial crossed product for the partial automorphism δe of the coefficient C ∗ -algebra B. That is C ∗ (X, α) = B ⋊δe Z and for C ∗ (X, α) we shall also write C ∗ (A, δ). Remark 4.2. In the case α is injective, equivalently δ is a partial automorphism, the e are equal and the covariance algebra of (A, δ) is simply the systems (A, δ) and (B, δ) partial crossed product. In particular, if α is a full homeomorphism, equivalently δ is an automorphism, then C ∗ (A, δ) is the classic crossed product. As we shall see, in the case α is surjective, equivalently δ is a monomorphism, C ∗ (A, δ) is the crossed product by a monomorphism considered for instance in [22, 24, 12, 1], cf. Remark 4.10. P k 1 e Let N e n (X)} such k=−N ak u stands for the element a in L = {a ∈ l (Z, B) : a(n) ∈ C∆ that a(k) = ak for |k| 6 N, and a(k) = 0 otherwise. In view of the defined operations on L it is clear that u is a partial isometry, uk is u to power k and (uk )∗ = u−k , so this notation should not cause any confusion. Using the natural injection B ∋ a 7→ au0 ∈ L e with the subalgebra of C ∗ (A, δ), see [10, Corollary 3.10]. Recalling we identify B = C(X) the identification from Definition 3.6 we have A ⊂ B ⊂ C ∗ (A, δ) = B ⋊θ Z.

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Example 4.3. The covariance algebra of the dynamical system considered in Examples 2.16 and 3.9 is the Toeplitz algebra. Indeed, the coefficient algebra B consists of convergent and δe is a forward shift so the partial crossed product B ⋊δe Z, cf. [10], is the Toeplitz algebra. Example 4.4. Let us go back again to Example 2.7 (see also Example 3.16). It is immediate to see that C ∗ (X, α) = C ∗ (X ′ , α′ ) and invoking [10], or [20, Example 2.5] we can identify this algebra with M3 ⊕M3 where M3 is the algebra of complex matrices 3×3. If we set A = C(X) and A′ = C(X ′ ) then due to the above remark we note that A and A′ consist of the matrices of the form 







 ax0 0 0   ax0 0 0       0 a ⊕ 0 a , 0 0 x1 x1         0 0 ax2 0 0 ay2

and









 ax0 0 0   ax0 0 0       0 a ⊕ 0 a 0  0 x1 y1         0 0 ax2 0 0 ay2

′ The  dynamics   on A and A are implemented by the partial isometry 0 0 0 0     1 0 0 . The coefficient algebras B0 = {S∞ U ∗n (A)U n } and ⊕ 0 n=0       0 010    axe0 0 0    S ∗n ′ n  0 a ⊕ B0′ = { ∞ U (A )U } equal with the algebra of diagonal matrices 0 n=0 x e1     0 0 axe2    aye0 0 0     0 a 0  , and it is straightforward that M3 ⊕ M3 = C ∗ (A, U) = C ∗ (A′ , U) and ye1     0 0 aye2 A ⊂ A′ . In fact A is the smallest C ∗ -subalgebra of M3 ⊕ M3 such that U ∗ U ∈ A, U(·)U ∗ is an endomorphism of A and C ∗ (A, U) = M3 ⊕ M3 .

respectively.  0 0  U :=  1 0  01

Example 4.5. It is known, cf. [10, 20], that an arbitrary finite dimensional C ∗ -algebra can be expressed as a covariance algebra (partial crossed product) of a certain dynamical system. The foregoing example inspires us to present the smallest such system in the sense that space X has the least number of points. Let A = Mn1 ⊕ ... ⊕ Mnk where 1 6 n1 6 ... 6 nk be a finite dimensional C ∗ -algebra and let us first assume that is there is no factor M1 in the decomposition of A, that is n1 > 1. We set X = {x1 , x2 , ..., xnk −1 , yn1 , yn2 , ..., ynk }, so |X| = nk + k − 1, and α(xm ) = xm−1 , for m = 2, ..., nk − 1; α(ynm ) = xnm −1 , for m = 1, ..., k,

B.K. Kwa´sniewski / Central European Journal of Mathematics 3(4) 2005 718–765

*    q

x q nk−1 - . . .

- x q nm −1 *   q

y nk

y nm

- xq n1 −1 *    q

...

...

749 - xq 1

y n1

It is clear that C ∗ (X, α) = A, see [20, Example 2.5]. In order to include algebras containing a number, say l, of one-dimensional factors one should simply add l points to the above diagram. Example 4.6. Let a be the bilateral weighted shift on a separable Hilbert space H and let a have the closed range. We have the polar decomposition a = U|a|, where |a| is a diagonal operator and U is the bilateral shift. If we denote by A the commutative C ∗ -algebra C ∗ (1, {U n |a|U ∗n }n∈N ), then δ(·) = U(·)U ∗ is a unital injective endomorphism of A and hence the dynamical system (X, α) corresponding to (A, δ) is such that α : e has the form X → X is onto. It is immediate that the coefficient algebra B = C(X) ∗ n ∗n ∗ C (1, {U |a|U }n∈Z ). Thus due to [23, Theorem 2.2.1], C (a) = B ⋊δe Z and so C ∗ (a) = C ∗ (X, α).

Following [23] we present now the canonical form of (X, α). Let Y denote the spectrum Q n of |a| and let T : X → ∞ n=0 Y be defined by T (x) = (x(|a|), x(δ(|a|)), ..., x(δ (|a|)), ...). Then similarly to [23] we infer that T is a homeomorphism of X onto T (X), where T (X) is Q given the topology induced by the product topology on ∞ n=0 Y , and under T , α becomes a shift on the product space T (X).

4.2 Universality of C ∗(X, α) Now we are in position to prove the main result of this section which justifies the anticipating Definition 4.1. We shall base the proof on the results from the previous section and some known facts concerning the partial crossed product [10]. We adopt the commonly used notation U −n = U ∗n where U is a partial isometry and n ∈ N. Theorem 4.7. Let (π, U, H) ∈ CovFaithRep (A, δ) or (π, U, H) ∈ CovRep (A, δ) in the case δ is a monomorphism. Then the formula (π × U)( (n)

(n)

N X

n=−N

n

a(n)u ) =

N ∞ X X

n=−N

k=0

 (n) U ∗k π(ak )U k U n

(30)

(n)

where a(n) = [a0 , a1 , ..., ak , ..] ∈ B0 , establishes an epimorphism of the covariance algebra C ∗ (A, δ) onto the C ∗ -algebra C ∗ (π(A), U) generated by π(A) and U. Proof. In both cases, (π, U, H) ∈ CovFaithRep (A, δ) or (π, U, H) ∈ CovRep (A, δ) and e see δ is injective, (π, U, H) extends to the covariant representation (π, U, H) of (B, δ), ∗ Theorems 3.10, 3.14. Since C (π(A), U) = B ⋊δe Z we obtain, by [10, Proposition 5.5],

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that (π × U)(

N X

n=−N

n

a(n)u ) =

N X

π(a(n))U n .

n=−N

establishes the representation of C ∗ (A, δ), and by Theorems 3.10 and 3.14, (π × U) is in fact given by (30).  Due to Definition 3.1, Corollary 3.13, and the preceding Theorem 4.7 we can alternatively define the covariance algebra to be the universal unital C ∗ -algebra generated by a copy of A and a partial isometry u subject to relations u∗ u ∈ A,

δ(a) = uau∗ ,

a ∈ A,

see also Proposition 1.6. In particular the above relations imply that uu∗ = P∆1 and u∗ u = P∆−1 . Thus δ is injective iff u is isometry, and δ is an automorphism iff u is unitary. Theorem 4.8. Let σ be a representation of C ∗ (A, δ) on a Hilbert space H. Let π denote the restriction of σ onto A and let U = σ(u). Then (π, U, H) ∈ CovRep (A, δ). e where π is σ restricted Proof. By [10, Theorem 5.6] we have (π, U, H) ∈ CovRep (B, δ) to B. Hence by Theorem 3.11 we get (π, U, H) ∈ CovRep (A, δ).  Corollary 4.9. If δ is a monomorphism then the correspondance (π, U, H) ←→ (π × U), cf. Theorem 4.7, is a bijection between CovRep (A, δ) and the set of all representations of C ∗ (A, δ). Proof. In virtue of Theorem 4.7 the mapping (π, U, H) ←→ (π × U) is a well defined injection and by Theorem 4.8 it is also a surjection.  Remark 4.10. Corollary 4.9 can be considered as a special case of Theorem 2.3 from the paper [22] where (twisted) crossed products by injective endomorphisms were investigated. However, our approach is slightly different. Oversimplifying; Murphy defines the algebra C ∗ (A, δ) as pZp where Z is the full crossed product of direct limit B = lim −−→A and p is a certain projection from B, whereas we include the projection p in the direct limit B=− lim −→An ⊂ B, see Proposition 3.7, and hence take the partial crossed product.

5

Invariant subsets and the topological freeness of partial mappings

The present section is devoted to the generalization of two important notions of the theory of crossed products. We start with a definition of α-invariant sets. With help of this notion we will describe (in the next section) the ideal structure of the covariance algebra. Next we introduce a definition of topological freeness - a property which is a

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powerful instrument when used to construct faithful representations of covariance algebra. Thanks to that in Section 6 we prove a version of the Isomorphism Theorem.

5.1 Definition of the invariance under α and α e and their interrelationship

The Definition 5.1 to follow might look strange at first however the author’s impression is that among the others this one is the most natural generalization of that from [13, Definition 2.7] for the case considered here. One may treat α-invariance as the invariance under the partial action of N on X. Definition 5.1. Let α be a partial mapping of X. A subset V of X is said to be invariant under the partial mapping α, or shorter α-invariant, if αn (V ∩ ∆n ) = V ∩ ∆−n ,

n = 0, 1, 2... .

(31)

When α is injective then we have another mapping α−1 : ∆−1 → X and life is a bit easier. Proposition 5.2. Let α be an injective partial mapping and let V ⊂ X. Then V is α-invariant if and only if one of the following conditions holds i) V is α−1 -invariant ii) for each n = 0, ±1, ±2, ..., we have αn (V ∩ ∆n ) ⊂ V iii) α(V ∩ ∆1 ) ⊂ V and α−1 (V ∩ ∆−1 ) ⊂ V , iv) α(V ∩ ∆1 ) = V ∩ ∆−1 . Proof. The equivalence of invariance of V under α and α−1 is straightforward, and so are implications i)⇒ ii)⇒ iii). To prove iii) ⇒ iv) let us observe that since α(V ∩ ∆1 ) ⊂ ∆−1 and α−1 (V ∩ ∆−1 ) ⊂ ∆1 we get α(V ∩ ∆1 ) ⊂ V ∩ ∆−1 and α−1 (V ∩ ∆−1 ) ⊂ V ∩ ∆1 . The latter relation implies that V ∩ ∆−1 = α(α−1 (V ∩ ∆−1 )) ⊂ α(V ∩ ∆1 ) and so α(V ∩ ∆1 ) = V ∩ ∆−1 . The only thing left to be shown is that iv) implies α-invariance of V . We prove this by induction. Let us assume that αk (V ∩∆k ) = V ∩∆−k , for k = 0, 1, ..., n−1 . By injectivity it is equivalent to α−k (V ∩ ∆−k ) = V ∩ ∆k , for k = 0, 1, ..., n − 1 . As ∆n ⊂ ∆n−1 and ∆−n ⊂ ∆−(n−1) we have αn (V ∩ ∆n ) ⊂ V ∩ ∆−n and α−n (V ∩ ∆−n ) ⊂ V ∩ ∆n . Applying αn to the latter relation we get V ∩ ∆−n ⊂ αn (V ∩ ∆n ) and hence αn (V ∩ ∆n ) = V ∩ ∆−n .  Item ii) tells us that Definition 5.1 extends [13, Definition 2.7] in the case of a single partial mapping. Let us note that if ∆1 6= X and α is not injective, then none of items ii)-iv) is equivalent to (31) and therefore none of them could be used as a definition of α-invariance. Example 5.3. Indeed, let X = {x0 , x1 , x2 , y2, y3 } and let α be defined by the relations

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α(y3 ) = y2 , α(y2 ) = α(x2 ) = x1 and α(x1 ) = x0 : q

x0

q  y2   ) qi PP x1P P q

q

y3

x2 Then the set V1 = {x0 , x1 , x2 } fulfills item iv) but it is not invariant under α in the sense of Definition 5.1. Whereas the set V2 = {x0 , x1 , y2 , y3} is α-invariant but it does not fulfill items ii) and iii). We shall show that in general there are less α-invariant sets in X than α e-invariant sets e in X, cf. Theorem 5.5 and a remark below, but fortunately there is a natural one-to-one correspondence between closed sets invariant under α and closed sets invariant under α e, see Theorem 5.7. We start with a lemma. Lemma 5.4. Let α be a partial mapping of X and let U ⊂ X be invariant under α. Then we have αk (∆n+k ∩ U) = U ∩ ∆n ∩ ∆−k ,

n, k = 0, 1, 2, ... .

Proof. As αk (U ∩ ∆k ) = U ∩ ∆−k and U ∩ ∆n+k ⊂ U ∩ ∆k we have αk (∆n+k ∩ U) ⊂ U ∩ ∆n ∩ ∆−k , for k, n ∈ N. On the other hand, for every x ∈ U ∩ ∆n ∩ ∆−k there exists y ∈ U ∩ ∆k such that αk (y) = x. Since x ∈ ∆n we we have y ∈ U ∩ ∆n+k and hence U ∩ ∆n ∩ ∆−k ⊂ αk (∆n+k ∩ U).  e be its reversible Theorem 5.5. Let (α, X) be a partial dynamical system and let (e α, X) e → X be the projection defined by (17). Then the map extension. Let Φ : X e⊃U e −→ U = Φ(U e) ⊂ X X

(32)

e onto the family of α-invariant is a surjection from the family of α e-invariant subsets of X e (see Definition 2.13)), then subsets of X. Furthermore if {Un }n∈N are the sections of U e is α U e-invariant if and only if U0 is α-invariant and Un = U0 ∩ ∆n ,

n = 0, 1, 2, ... .

e be α e ). Then by (19) and α e , for Proof. Let U e-invariant and let U = Φ(U e-invariance of U each n ∈ N, we have e ∩∆ e n )) = Φ(U e ∩∆ e −n ) = U ∩ ∆−n αn (U ∩ ∆n ) = Φ(e α n (U

and hence U is invariant under α and the mapping (32) is well defined. Moreover, if Un , e , then by invariance of U e under α n ∈ N, are the sections of U e−1 (see Proposition 5.2) we get e ∩∆ e −n )) = Φ(U e ∩∆ e n ) = U ∩ ∆n Un = Φ(e α−n (U

where U = U0 is α-invariant. Now, we show that the mapping (32) is surjective. Let U be any nonempty set invariant

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e of the form under α and let us consider U   e := U × (U ∩ ∆1 ∪ {0}) × ... × (U ∩ ∆n ∪ {0}) × ... ∩ X. e U

e ) = U (note that in general U e may occur to be empty). What we need to prove is that Φ(U e ) ⊂ U. In order to prove that U ⊂ Φ(U e ) we fix an arbitrary point It is clear that Φ(U e such that Φ(e x0 ∈ U and suppose that there does not exist x e in U x) = x0 . We will e and thereby obtain a contradiction. construct an infinite sequence (x0 , x1 , x2 , ...) in U e. Indeed, we must have x0 ∈ U ∩ ∆−1 , for otherwise we can take x e = (x0 , 0, 0, ...) ∈ U Hence by Lemma 5.4 there exists x1 ∈ U ∩ ∆1 such that α(x1 ) = x0 . Suppose now we have chosen n−1 points x1 , ..., xn such that xk ∈ U ∩∆k and α(xk ) = xk−1 for k = 1, ..., n, e. then xn must be in U ∩ ∆n ∩ ∆−1 , for otherwise we can take x e = (x0 , x1 , ..., xn , 0, ...) ∈ U Hence by Lemma 5.4 there exists xn+1 ∈ U ∩ ∆n+1 such that α(xn+1 ) = xn . This ensure us that there is a sequence x e = (x0 , x1 , x2 , ...) such that xn ∈ U ∩ ∆n and α(xn ) = xn−1 , e and we arrive at the contradiction. for all n = 1, 2, ... . Thus x e∈U e it suffices By virtue of item v) in Proposition 5.2 in order to prove the α e-invariance of U to show that e ∩∆ e 1) ⊂ U e e ∩∆ e −1 ) ⊂ U e α e (U and α e−1 (U e α but this follows immediately from the form of U, e, α e−1 and from α-invariance of U. Thus according to the first part of the proof we conclude that, for each n ∈ N, the n-section e is equal to U ∩ ∆n . The proof is complete. Un of U 

T Corollary 5.6. If α is injective on the inverse image of ∆−∞ = n∈N ∆−n (for x ∈ ∆−∞ we have |α−1 (x)| = 1) then the mapping (32) from the previous theorem is a bijection. Proof. It suffices to apply Proposition 2.14.



Under the hypotheses of Theorem 5.5, surjection considered there might not be a bijection. For instance in Example 2.16 we have three α-invariant sets: X, {x0 }, ∅, and four α e-invariant sets: N, N, ∞, ∅. However the mapping (32) is always bijective when restricted to closed invariant sets. Theorem 5.7. The mapping (32) is a bijection from the family of α e-invariant closed sets onto the family of α-invariant closed sets. e such that Φ(U e) = Proof. By Theorems 5.5 and 2.15, for every α e-invariant closed subset U U we have   e = U × (U ∩ ∆1 ∪ {0}) × ... × (U ∩ ∆n ∪ {0}) × ... ∩ X, e U (33)

e is uniquely determined by U. Since X e and X are compact, and Φ : X e →X that means U is continuous the set U is closed. Hence Φ maps injectively the family of closed α e-invariant sets into the family of closed α-invariant sets.

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On the other hand, if U is closed and α-invariant then by definition of the product e given by (33) is also closed and according to Theorem 5.5, Φ(U e ) = U. topology, the set U Thus the proof is complete.  The next important notion which we shall need to obtain a simplicity criteria for covariance algebra (see Corollary 6.6) is the notion of minimality, cf. [13]. Definition 5.8. A partial continuous mapping α (or a partial dynamical system (X, α)) is said to be minimal if there are no α-invariant closed subsets of X other than ∅ and X. Proposition 5.9. A partial dynamical system (X, α) is minimal if and only if its ree α versible extension (X, e) is minimal.

Proof. An easy consequence of Theorem 5.7.



When α is injective, the binary operations ” ∪” and ” ∩”, or equivalently partial order relation ” ⊂ ”, define the lattice structure on the family of α-invariant sets, see [13]. The situation changes when α is not injective. Of course ” ⊂ ” is still a partial order relation which determines the lattice structure, but it may happen that the intersection of two α-invariant sets is no longer α-invariant. Example 5.10. Let (X, α) and (X ′ , α′ ) be dynamical systems from Example 2.7. It is easy to verify that there are four sets invariant under α: X, V1 = {x0 , x1 , x2 }, V2 = {x0 , x1 , y2 } and ∅; and four sets invariant under α′ : X ′ , V1′ = {x′0 , x′1 , x′2 }, V2′ = {x′0 , y1′ , y2′ }, ∅. Hence neither V1 ∩ V2 nor V1′ ∩ V2′ is invariant. However there are four e Ve1 = {e invariant subsets: X, x0 , x e1 , x e2 }, Ve2 = {e y0 , ye1, ye2 }, ∅ on the reversible extension ′ ′ e α e ,α level ((X, e) = (X e )) and Ve1 ∩ Ve2 = ∅ is invariant of course. Definition 5.11. We denote by closα (X) the lattice of α-invariant closed subsets of X where the lattice structure is defined by the partial order relation ” ⊂ ”.

e ∼ According to Theorem 5.7, Φ determines the lattice isomorphism closαe (X) = closα (X).

5.2 Topological freeness Recall now that a partial action of a group G on a topological space X is said to be topologically free if the set of fixed points Ft , for each partial homeomorphism αt with t 6= e, has an empty interior [13]. In view of that, the next definition constitutes a generalization of topological freeness notion to the class of systems where dynamics are implemented by one, not necessarily injective, partial mapping. Definition 5.12. Let α : ∆ → X be a continuous partial mapping of Hausdorff’s topological space X. For each n > 0, we set Fn = {x ∈ ∆n : αn (x) = x}. It is said that the

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action of α (or briefly α) is topologically free, if every open nonempty subset U ⊂ Fn possess ’an exit’, that is there exists a point x ∈ U such that one of the equivalent conditions hold i) for some k = 1, 2, ..., n we have |(α−k (x))| > 1, ii) for some k = 1, 2, ..., n α−1 (αk (x)) 6= {αk−1(x)}. We supply now some characteristics of this topological freeness notion. Proposition 5.13. The following conditions are equivalent i) α is topologically free, ii) for each n > 0 and every open nonempty subset U ⊂ Fn there exist points x ∈ U, y ∈ ∆1 \ Fn and a number k = 1, 2, ..., n, such that α(y) = αk (x). iii) for each n > 0, the set {x ∈ ∆n−1 : αk−n (x) = {αk (x)} for k = 0, 1, ..., n − 1} has an empty interior. Proof. i) ⇒ ii). Let U ⊂ Fn be an open nonempty set. Let x ∈ U and k = 1, ..., n be such that item ii) from Definition 5.12 holds. We take y ∈ α−1 (αk (x)) such that y 6= αk−1 (x). Then α(y) = αk (x) and since αn (y) = αk−1(x) we have y ∈ / Fn . ii) ⇒ iii). Suppose that for some n > 0 there exists a nonempty open subset U of {x ∈ ∆n−1 : αk−n (x) = {αk (x)} for k = 0, 1, ..., n − 1}. It is clear that U ⊂ Fn and hence for some k0 = 1, 2, ..., n, there exists y ∈ ∆1 \ Fn such that α(y) = αk0 (x). Taking k = k0 − 1 we obtain that y ∈ αk−n (x) = {αk (x)} and thus we arrive at a contradiction since αk (x) ∈ Fn and y ∈ / Fn . iii) ⇒ i). Suppose that α is not topologically free. Then there exists an open nonempty set U ⊂ Fn such that for all x ∈ U and k = 1, ..., n, we have |α−k (x)| = 1. It is not hard to see that U ⊂ {x ∈ ∆n−1 : αk−n (x) = {αk (x)} for k = 0, 1, ..., n − 1} and thereby we arrive at the contradiction.  The role similar to the one which topological freeness plays in the theory of crossed products is the role played in the theory of C ∗ -algebras associated with graphs by the condition that every circuit in a graph has an exit (see, for example, [5, 11]). The connection between these two properties is not only of theoretical character, see for instance [11, Proposition 12.2]). Taking this into account the following two simple examples might be of interest. Before that let us establish the indispensable notation. Let A = (A(i, j))i,j∈{1,...,N } be the matrix with entries in {0, 1}. It can be regarded as an adjacency matrix of a directed graph Gr(A): the vertices of Gr(A) are numbers 1,...,N and edges are pairs (x, y) of vertices such that A(x, y) = 1. By a path in Gr(A) we mean a sequence (x0 , x1 , ...xn ) of vertices such that A(xk , xk+1 ) = 1 for all k. A circuit, or a

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loop, is a finite path (x0 , ..., xn ) such that A(xn , x0 ) = 1. Finally a circuit (x0 , x1 , ..., xn ) is said to have an exit if, for some k, there exists y ∈ {1, ..., N} with A(xk , y) = 1 and y 6= xk+1 (mod n) . Example 5.14. Let (X, α) be a partial dynamical system such that X = {1, ..., N} is finite. If we define A by the relation: A(x, y) = 1 iff α(y) = x, then α is topologically free if and only if every loop in Gr(A) has an exit. This is an easy consequence of the fact that Gr(A) is the graph of the partial mapping α with reversed edges. We shall say that a circuit (x0 , x1 , ..., xn ) has an entry if, for some k, there exists y ∈ {1, ..., N} with A(y, xk ) = 1 and y 6= xk−1 (mod n). Example 5.15. Let (XA , σA ) be a dynamical system where σA is a one-sided Markov subshift associated with a matrix A, see page 733. One-sided subshift σA acts topologically free if and only if every circuit in Gr(A) has an exit or an entry. Indeed, if there exists a loop (y0 , ..., yn ) in Gr(A) which has no exit and no entry then U = {(xk )k∈N ∈ XA : x0 = y0 } = {(y0, y1 , ..., yn , y0 , ...)} is an open singleton, U ⊂ Fn+1 and U has no ’exit’ in the sense of Definition 5.12, thereby σA is not topologically free. On the other hand, if every loop in Gr(A) has an exit or an entry, and if x = (x1 , ..., xn , x1 , ...) is an element of an open subset U ⊂ Fn for some n > 0, then the loop (x1 , ..., xn ) must have an entry, as it clearly has no exit. Hence (x1 , ..., xn ) we have |(σA−k (x))| > 1, for some k = 1, 2, ..., n, and thus σA is topologically free. We end this section with the result which, in a sense, justifies Definition 5.12, and is the main tool used to prove the Isomorphism Theorem. en : α Theorem 5.16. Let Fn = {x ∈ ∆n : αn (x) = x} and Fen = {e x ∈ ∆ en (e x) = x e}, n ∈ N \ {0}. We have e : xk ∈ Fn , k ∈ N}, Fen = {(x0 , x1 , ...) ∈ X

n = 1, 2, ...,

(34)

and α is topologically free if and only if α e is topologically free.

Proof. Throughout the proof we fix an n > 0. It is clear that Fn and Fen are invariant under α and α e respectively (see Definition 5.1), and that Φ(Fen ) = Fn . By virtue of Theorem 5.5 we have   e Fen = Fn × (Fn ∩ ∆1 ∪ {0}) × ... × (Fn ∩ ∆k ∪ {0}) × ... ∩ X.

T e and hence (34) But, since Fn ⊂ k∈Z ∆k we obtain Fen = (Fn × Fn × ... × Fn × ...) ∩ X holds. Now suppose that α is topologically free and on the contrary that there exists an open e ⊂ Fen . Without loss of generality, we can assume that it has the form nonempty subset U   e e U = U0 × U1 ... × Um × ∆m+1 ∪ {0} × ∆m+2 ∪ {0} × ... ∩ X

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e ⊂ Fen = (Fn × Fn × ... × Fn × ...) ∩ X, e where U0 , U1 , ..., Um are open subsets of X, and as U they are in fact subsets of Fn , and it is readily checked that m   \ −k e e U = Fn × Fn ... × α (Um−k ) × ∆m+1 ∪ {0} × ∆m+2 ∪ {0} × ... ∩ X. k=0

T −k The set U := m k=0 α (Um−k ) is an open and nonempty subset of Fn . Hence, due to the topological freeness of α there exists y ∈ / Fn and k = 1, ..., n, such that α(y) = αk (x) for e such that xm := x, xm+i := αn−i(x) some x ∈ U. Taking any element x e = (x0 , x1 , ...) ∈ X e and for i = 1, ..., n − k, and xm+n−k+1 = y we arrive at the contradiction, because x e∈U x e∈ / Fen . Finally suppose that α is not topologically free. Then there exists an open nonempty subset U ⊂ Fn such that, for all x ∈ U, |α−k (x)| = 1 and so e = {(x, αn−1 (x), αn−2(x), ..., α1 (x), x, αn−1(x), ...) ∈ X e : x ∈ U} = (U×(X∪{0})×...)∩X e U is an open nonempty subset of Fen . Hence α e is not topologically free and the proof is complete. 

6

Ideal structure of covariance algebra and the Isomorphism Theorem

It is well-known that every closed ideal of A = C(X) is of the form CU (X) where U ⊂ X is open, and therefore we have an order preserving bijection between open sets and ideals. The Theorem 3.5 from [13] can be regarded as a generalization of this fact; it says that, under some assumptions, there exists a lattice isomorphism between open invariant sets and ideals of the partial crossed product. In this section we shall prove the new useful variant of this theorem. The novelty is that in our approach (cf. Theorem 5.7) it is more natural to investigate a correspondence between ideals of the covariance algebra and closed invariant sets. After that we shall prove the main result of this paper, a version of the Isomorphism Theorem where the main achievement is that we do not assume any kind of reversibility of an action on a spectrum of a C ∗ -dynamical system.

6.1 Lattice isomorphism of closed α-invariant sets onto ideals of C ∗(X, α) Let us start with the proposition which is an attempt of describing the concept of invariance on the algebraic level, cf. [13, Definition 2.7]. For that purpose we will abuse notation concerning endomorphism δ and denote by δ n , n ∈ N, morphisms δ n : C(∆−n ) → C(∆n ) of composition with αn : ∆n → ∆−n . We believe that this notation does not cause confusion, although we stress that set ∆−n does not have to be open and hence we can not identify C(∆−n ) with a subset of C(X). For instance, it may happen that ∆−n is not empty but have an empty interior, and then C∆−n (X) is empty while C(∆−n ) is not. We will also

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abuse notation concerning subsets and write B∩C(∆−n ) for {a restricted to ∆−n : a ∈ B} where B ⊂ C(X). Proposition 6.1. Let V be a closed subset of X and let I = CX\V (X) be the corresponding ideal. Then for n ∈ N we have i) αn (V ∩ ∆n ) ⊂ V ∩ ∆−n iff a ∈ I ∩ C(∆−n ) =⇒ δ n (a) ∈ I ∩ C(∆n ), ii) αn (V ∩ ∆n ) ⊃ V ∩ ∆−n iff δ n (a) ∈ I =⇒ a ∈ I, for all a ∈ C(∆−n ). Hence V is α-invariant (V ∈ closα (X)) if and only if ∀n∈N ∀a∈C(∆−n ) a ∈ I ∩ C(∆−n ) ⇐⇒ δ n (a) ∈ I ∩ C(∆n ).

(35)

Proof. i). Let αn (V ∩∆n ) ⊂ V ∩∆−n and let a ∈ I∩C(∆−n ) be fixed. Then for x ∈ V ∩∆n we have αn (x) ∈ V ∩ ∆−n , whence δ n (a) = a(αn (x)) = 0 and δ n (a) ∈ I ∩ C(∆n ). Now suppose αn (V ∩ ∆n ) * V ∩ ∆−n . Then there exists x0 ∈ V ∩ ∆n such that αn (x0 ) ∈ / n V ∩ ∆−n . As α (x0 ) ∈ ∆−n and V is closed, by Urysohn’s lemma, there is a function a0 ∈ C(X) such that a0 (αn (x0 )) = 1 and a0 (x) = 0 for all x ∈ V . Thus taking a to be the restriction of a0 to ∆−n we obtain a ∈ I ∩ C(∆−n ) but δ n (a)(x0 ) = 1, whence δ n (a) ∈ / I ∩ C(∆n ). ii). Let αn (V ∩ ∆n ) ⊃ V ∩ ∆−n and let a ∈ C(∆−n ) be such that δ n (a) ∈ I ∩ C(∆n ). Suppose on the contrary that a ∈ / I ∩ C(∆−n ). Then a(y0 ) 6= 0 for some y0 ∈ V ∩ ∆−n . Taking x0 ∈ V ∩ ∆n such that y0 = αn (x0 ) we arrive at the contradiction with δ n (a) ∈ I ∩ C(∆n ) because δ n (a)(x0 ) = a(y0 ) 6= 0. If αn (V ∩ ∆n ) + V ∩ ∆−n , then there exists x0 ∈ V ∩ ∆−n \ αn (V ∩ ∆n ). Similarly as in the proof of item i), using Urysohn’s lemma we can take a0 ∈ C(X) such that a0 (x0 ) = 1 and a0 |αn (V ∩∆n ) ≡ 0. Hence putting a = a0 |∆−1 we have a ∈ / I ∩ C(∆−n ) and n δ (a) ∈ I ∩ C(∆n ). In view of i) and ii), α-invariance of V is evidently equivalent to the condition (35).  Definition 6.2. If I is a closed ideal of A satisfying (35) then we say that I is invariant under the endomorphism δ, or briefly δ-invariant. In virtue of Proposition 6.1 it is clear that I is a δ-invariant ideal iff I = CX\V (X) where V is a closed α-invariant set. Thus, using Theorem 5.7 one can obtain a correspondence between the invariant ideals of A and invariant ideals of B. To this end we denote e by hIiB, δe the smallest δ-invariant ideal of B containing I.

e be Proposition 6.3. Let I = CX\V (X) be a δ-invariant ideal of A and let Ve ∈ closαe (X) such that Φ(Ve ) = V . Then e hCX\V (X)iB, δe = CX\ e Ve (X),

and the mapping I 7−→ hIiB, δe establishes an order preserving bijection between the e family of δ-invariant ideals of A and δ-invariant ideals of B. Moreover, the inverse of the mentioned bijection has the form Ie 7−→ Ie ∩ A.

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Proof. In order to prove the first part of the statement we show that the support [ e : f (e S= {e x∈X x) 6= 0} f ∈hCX\V (X)i

e B, δ

e \ Ve . of the ideal hCX\V (X)iB, δe is equal to X Let a ∈ I = CX\V (X). We identify a with [a, 0, ...] ∈ B and since x0 ∈ V for any e x e = (x0 , ...) ∈ Ve , we note that [a, 0, ...](e x) = a(x0 ) = 0, that is a = [a, 0, ...] ∈ CX\ e Ve (X). e e is δ-invariant, e whence S ⊂ X e \ Ve . As C e e (X) we get hCX\V (X)i ⊂ C e e (X), X\V

e B, δ

X\V

e \ Ve . The form of Ve (compare Theorem 5.5) implies Now, let x e = (x0 , ..., xk , ...) ∈ X that there exists n ∈ N such that xn ∈ / V . According to Urysohn’s lemma there exists a ∈ CX\V (X) such that a(xn ) = 1. By invariance, all the elements δek (a) and δe∗k (a) for k ∈ N, belong to hIiB, δe . In particular δe∗n (a) = [0, ..., aδ n (1), 0, ...] ∈ hIiB, δe where e \ Ve = S. δe∗ (a)(e x) = a(xn ) = 1 6= 0. Thus x e ∈ S and we get X In virtue of Theorem 5.7 the relation Φ(Ve ) = V establishes an order preserving bijection e and closα (X) hence the relation hCX\V (X)i = C e e (X) e establishes between closαe (X) e B, δ

X\V

e such a bijection too. The inverse relation CX\ e Ve (X) ∩ A = CX\V (X) is straightforward.

Let us recall that we identify B with a subalgebra of the covariance algebra C ∗ (A, δ). Therefore for any subset K of B we denote by hKi an ideal of C ∗ (A, δ) generated by K. The next statement follows from the preceding proposition and Theorem 3.5 from [13]. Theorem 6.4. Let (A, δ) be a C ∗ -dynamical system such that α has no periodic points. Then the map V 7−→ hCX\V (X)i

is a lattice anti-isomorphism from closα (X) onto the lattice of ideals in C ∗ (A, δ). Moree such that Φ(Ve ) = V the following relations hold over, for Ve ∈ closαe (X) e hCX\V (X)i = hCX\ e Ve (X)i,

e hCX\V (X)i ∩ B = CX\ e Ve (X),

hCX\V (X)i ∩ A = CX\V (X).

Proof. Since α has no periodic points neither does its reversible extension α e. The co∗ e ⋊e Z and Z is an amenable variance algebra C (A, δ) is the partial crossed product C(X) δ e has the approximation property, see [13]. Thus in view of Theoe δ) group. Hence (C(X), e is a lattice anti-isomorphism from closαe (X) e rem 3.5 from [13], the map Ve 7−→ hC e e (X)i X\V

∗

e e onto the lattice of ideals of C (A, δ), and the inverse relation is hCX\ e Ve (X)i ∩ C(X) = e C e e (X). X\V

e and V ∈ closα (X) be such that Φ(Ve ) = V . We show that Now, let Ve ∈ closαe (X) e hCX\V (X)i = hCX\ e Ve (X)i. e e e On one hand, by Proposition 6.3 we have hCX\ e Ve (X)i ∩ C(X) ∩ A = e Ve (X)i ∩ A = hCX\ e e ∩ A = CX\V (X) and hence hCX\V (X)i ⊂ hC e e (X)i. On the other hand, C e e (X) X\V

X\V

e e is a δ-invariant e containing CX\V (X), and so it also hCX\V (X)i ∩ C(X) ideal of C(X) e e contains hCX\V (X)iB, δe = CX\ e Ve (X)i ⊂ hCX\V (X)i. e Ve (X). Hence hCX\

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Concluding, we have the lattice isomorphism between sets V ∈ closα (X) and Ve ∈ e e and closαe (X), and the lattice anti-isomorphism between sets Ve ∈ closαe (X) e ideals hCX\V (X)i = hCX\  e Ve (X)i, hence V 7−→ hCX\V (X)i is anti-isomorphism. Example 6.5. In Example 4.5 the only α-invariant sets are {x1 , ..., xm−1 , ym }, m = 1, ..., k, and their sums. The corresponding ideals are Mnm , m = 1, ..., k, and their direct sums.

We automatically get a simplicity criteria for the covariance algebra. We say that (X, α) forms a cycle, if X = {x0 , ..., xn−1 } is finite and α(xk ) = xk+1(mod n) , k = 0, ...n − 1. Corollary 6.6. Let α be minimal. If (X, α) does not form a cycle then C ∗ (A, δ) is simple. Proof. It suffices to observe that if α is minimal then α has no periodic points or (X, α) forms a cycle. Hence we can apply Theorem 6.4.  Example 6.7. If (X, α) does form a cycle then there are infinitely many ideals in C ∗ (A, δ). Indeed if we have A = Cn and δ(x1 , ..., xn ) = (xn , x1 , ..., xn−1 ), it is known that the partial crossed product Cn ⋊δ Zn is isomorphic to the algebra Mn of complex matrices n × n and hence C ∗ (A, δ) ֒→ C(S 1 ) ⊗ Mn = C(S 1 , Mn ).

6.2 The Isomorphism Theorem The Isomorphism Theorem simply states that under some conditions epimorphism from Theorem 4.7 is in fact an isomorphism. We will prove here two statements of that kind, Theorems 6.9 and 6.11, in the literature however only the latter one is named the Isomorphism Theorem. A significant role in the proofs of both of these statements plays a certain inequality which ensures the existence of conditional expectation onto the coefficient algebra, and which appears in different versions in a number of sources concerning various crossed products. For references see [18, 19, 2, 3, 23], and for the greatest similarity with the following Definition 6.8 and Theorem 6.9 see [1, Theorem 1.2]. Definition 6.8. We say that a C ∗ -algebra C ∗ (C, U) generated by a C ∗ -algebra C and an element U possesses the property (∗) if the following inequality holds k

M X k=0

(0)

U ∗k π(ak )U k k ≤ k

N M X X

n=−N

k=0

 (n) U ∗k π(ak )U k U n k

(∗)

(n)

for any ak ∈ C and M, N ∈ N. Theorem 6.9. Let (π, U, H) ∈ CovFaithRep (A, δ). Then formula (30) establishes an

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isomorphism between the covariance algebra C ∗ (A, δ) and the C ∗ -algebra C ∗ (π(A), U) if and only if C ∗ (π(A), U) possess the property (∗). Proof. Necessity. It suffices to observe that C ∗ (A, δ) = C ∗ (A, u) possess the property (∗), and this follows immediately from the fact that partial crossed products satisfy the appropriate version of this property, see [19, Remark 2.1] and [20, Proposition 3.5]. Sufficiency. By Theorem 3.10, (π, U, H) extends to the covariant faithful representation (π, U, H) of the coefficient C ∗ -algebra B. This extended representation satisfies assumptions of [19, Theorem 3.1] and as Z is amenable (π, U, H) give rise to the desired isomorphism, see also [19, Remark 3.2]  Corollary 6.10. Let v ∈ A be a partial isometry such that uu∗ ≤ v ∗ v, vv ∗ where u is the universal partial isometry in C ∗ (A, δ). Then the mapping Λv (u) = vu,

Λv (a) = a,

a ∈ A,

extends to an automorphism of C ∗ (A, δ). In particular, taking v = λ1, λ ∈ S 1 , we have the action Λ of the unit circle S 1 on C ∗ (A, δ) for which the fixed points set is the coefficient C ∗ -algebra B. Proof. By the above theorem C ∗ (A, δ) = C ∗ (A, u) possesses the property (∗). Clearly, the same is true for C ∗ (A, vu). Since uu∗ ≤ v ∗ v we have u = v ∗ vu = v ∗ (vu) ∈ C ∗ (A, uv), whence C ∗ (A, δ) = C ∗ (A, uv), and furthermore (vu)∗ vu = u∗ v ∗ vu = u∗ u ∈ A. Since uu∗ ≤ vv ∗ we have (vu)a(vu)∗ = uau∗ vv ∗ = uau∗, that is the element vu generates the same endomorphism of A as u, and hence applying the preceding theorem we conclude that Λv extends to an automorphism of C ∗ (A, δ). The rest is straightforward.  Now, we are in position to prove our variant of the celebrated Isomorphism Theorem. Theorem 6.11 (Isomorphism Theorem). Let (A, δ) be such that α is topologically free. Then for every (π, U, H) ∈ CovFaithRep (A, δ) the algebra C ∗ (π(A), U) possess property (∗). In other words, for any two covariant faithful representations (π1 , U1 , H1 ) and (π2 , U2 , H2 ), the mapping U1 7−→ U2 ,

π1 (a) 7−→ π2 (a),

a ∈ A,

determines an isomorphism of C ∗ (π1 (A), U1) onto C ∗ (π2 (A), U2 ). Proof. Due to Theorem 5.16, the partial homeomorphism α e is topologically free and according to Theorem 3.10 representations π1 and π2 give rise to covariant representations e Thus it is enough to (π 1 , U1 , H1 ) and (π 2 , U2 , H2 ) of the partial dynamical system (B, δ). apply the Theorem 3.6 from [19].  Corollary 6.12. Let A act nondegenerately on a Hilbert space H, let δ(·) = U(·)U ∗ where U ∈ L(H) is a partial isometry such that U ∗ U ∈ A, and let the generated partial mapping α be topologically free. Then C ∗ (A, U) ∼ = C ∗ (A, δ).

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The above corollary allow us, in the presence of topological freeness, consider only abstract covariance algebras. However, in various concrete specification while using the method mentioned after Theorem 1.4, it may happen that the Isomorphism Theorem can be applied to systems (A, δ) such that ∆−1 is not open and α is not topologically free. Example 6.13. Let A and U be as in Example 1.5, then the associated system forms a cycle and therefore it is not topologically free. However after passing to algebra C = C ∗ (A, U ∗ U) we obtain the dynamical system (X ∪ {y}, α) (see Example 1.5) q ..  xn−1 .. ) - q q P . P y x0 PP - q . q q x1 which is topologically free, cf. Example 5.14. Hence, due to the Isomorphism Theorem C ∗ (A, U) ∼ = C ∗ (X ∪ {y}, α). In particular, if n = 1 then C ∗ (A, U) is the Toeplitz algebra, see Examples 2.16, 3.9 and 4.3. Example 6.14. Consider Hilbert spaces H1 = L2µ ([0, 1]) and H2 = L2µ (R+ ) where µ is the Lebesgue measure. We fix 0 < q < 1 and 0 < h < ∞. Let A1 ⊂ L(H1 ) consists of operators of multiplication by functions from C[0, 1] and let U1 act according to (U1 f )(x) = f (q · x), f ∈ H1 . Similarly, let elements of A2 ⊂ L(H2 ) act as operators of multiplication by functions which are continuous on R+ = [0, ∞) and have limit at infinity, and let U2 be the shift operator (U2 f )(x) = f (x+h), f ∈ H2 . Then the dynamical systems associated to C ∗ -dynamical systems (A1, U1 (·)U1∗ ) and (A2 , U2 (·)U2∗ ) are topologically conjugate but the images of the generated mappings are not open (compare with Example 2.4). Thus we can not apply the Theorem 6.11 in the form it is stated. Nevertheless, endomorphisms of bigger algebras C1 = C ∗ (A1 , U1∗ U1 ) and C2 = C ∗ (A2 , U2∗ U2 ) do generate dynamical systems q

q

y,

y  sq 0



q

q2

q

q

q

1

q

0

Uq

h

Nq

2h

q 

∞

satisfying the assumptions of the Isomorphism Theorem. These dynamical systems are topologically conjugate by a piecewise linear mapping φ which maps nh into q n , n ∈ N ∪ {∞}, and y ′ into y, that is φ(x) = q n


q−1 x+1−n(q−1) , h

for x ∈ [nh, nh+1),

and φ(∞) = 0, φ(y ′) = y.

Therefore, by the Isomorphism Theorem, the mapping A1 ∋ a 7→ a ◦ φ ∈ A2 , and U1 7→ U2 , establishes the expected isomorphism; C ∗ (A1 , U1 ) ∼ = C ∗ (A2 , U2 ). Lastly, we would like to present an example which shows how the results achieved in this paper clarify the situation mentioned in the example from which we have started the introduction.

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Example 6.15 (Solenoid). Let H = L2µ (R) where µ is the Lebesgue measure on R, and let A ⊂ L(H) consists of the operators of multiplication by periodic continuous functions with period 1, that is A ∼ = C(S 1 ). Set the unitary operator U ∈ L(H) by the formula √ (Uf )(x) = 2 f (2x). Then for each a(x) ∈ A, UaU ∗ is the operator of mulitplication by the periodic function 1 ∗ a(2x)  x  with period 2 , and U aU is the operator of multiplication by the periodic function a with period 2. Hence 2 UAU ∗ ⊂ A and U ∗ AU * A.

The endomorphism U(·)U ∗ generate on the spectrum of A the mapping α given by α(z) = S z 2 for z ∈ S 1 , and the spectrum of the algebra B generated by n∈N U ∗n AU n is the solenoid S: B ∼ = C(S), cf. Example 2.12. Further more α is topologically free and therefore we have C ∗ (A, U) ∼ = C ∗ (S 1 , α) = C(S) ⋊F Z where in the right hand side stands the standard crossed product of B = C(S) by the automorphism induced by the solenoid map F , see Example 2.12.

Summary In this paper we introduced crossed product-like realization of the universal algebra associated to ’almost’ arbitrary commutative C ∗ -dynamical system (A, δ). This new realization generalizes the known constructions for C ∗ -dynamical systems where dynamics is implemented by an automorphism or a monomorphism. The primary gain of this is that we are able to describe important characteristics of the investigated object in terms of the underlying topological (partial) dynamical system (X, α), the tool which until now was used successfully only in the case of a (partial) automorphism. Namely, we have described the ideal structure of covariance algebra by closed invariant subsets of X, in particular simplicity criteria is obtained. Moreover we have generalized the topological freeness, the condition under which all the covariant faithful representations of (A, δ) are algebraically equivalent, see the Isomorphism Theorem. For applications this is probably the most important result of the paper. The important novelty in our approach is that the construction of covariance algebra here consists of two independent steps. The advantage of this is that one may analyse covariance algebra on two levels. First, one may study the relationship between initial C ∗ -dynamical system and the one generated on its coefficient C ∗ -algebra, and then one may apply known statements and methods as the latter system is more accessible (generated mapping on the spectrum of coefficient algebra is bijective). We indicate that recently (see [12]) a notion of crossed-product of a C ∗ -algebra by an endomorphism (or even partial endomorphism, see [14]) has been introduced, a construction

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which depends also on the choice of transfer operator. This construction is especially well adapted to deal with morphisms which generate local homeomorphisms. In particular it was used to investigate Cuntz-Krieger algebras, cf. [12, 14, 11]. However it seems that in the case that α is not injective there does not exist a transfer operator such that the aforementioned crossed-product is isomorphic to covariance algebra considered here, and in the case that α is injective the transfer operator is trivial, that is, it is α−1 and thus it does not add anything new to the system.

Acknowledgment The author wishes to express his thanks to A.V. Lebedev for suggesting the problem and many stimulating conversations, to A.K. Kwa´sniewski for his active interest in the preparation of this paper, and also to D. Royer for pointing an error in an earlier version of Theorem 2.2.

References [1] S. Adji, M. Laca, M. Nilsen and I. Raeburn: ”Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups”, Proc. Amer. Math. Soc., Vol. 122(4), (1994), pp. 1133–1141. [2] A. Antonevich: Linear Functional Equations. Operator Approach, Operator Theory Advances and Applications, Vol. 83, Birkhauser Verlag, Basel, 1996. [3] A. Antonevich and A.V. Lebedev: Functional differential equations: I. C ∗ -theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 70, Longman Scientific & Technical, Harlow, Essex, England, 1994. [4] B. Blackadar: K-theory for Operator algebras, Springer-Verlag, New York, 1986. [5] T. Bates, D. Pask, I. Raeburn and W. Szyma´ nski: ”C*-algebras of row-finite graphs”, New York J. Math, Vol. 6, (2000), pp. 307–324. [6] O. Bratteli and D. W. Robinson: Operator algebras and Quantum Statistical Mechanics I,II, New York 1979, 1980. [7] M. Brin and G. Stuck: Introduction to Dynamical Systems, Cambridge University Press, 2003. [8] J. Cuntz: ”Simple C ∗ -algebras generated by isometries”, Commun. Math. Phys., Vol. 57, (1977), pp. 173–185. [9] J. Dixmier: C ∗ -algebres et leurs representations, Gauter-Villars, Paris, 1969. [10] R. Exel: ”Circle actions on C ∗ -algebras, partial automorphisms and generalized Pimsner-Voiculescu exact sequence”, J. Funct. Analysis, Vol. 122, (1994), pp. 361– 401. [11] R. Exel and M. Laca: ”Cuntz-Krieger algebras for infinite matrices”, J. Reine Angew. Math., Vol. 512, (1999), pp. 119–172, http://front.math.ucdavis.edu/funct-an/9712008. [12] R. Exel: ”A new look at the crossed-product of a C ∗ -algebra by an endomorphism”, Ergodic Theory Dynam. Systems, Vol 23, (2003), pp. 1733–1750, http://front.math.ucdavis.edu/math.OA/0012084.

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[13] R. Exel, M. Laca and J. Quigg: ”Partial dynamical systems and C ∗ -algebras generated by partial isometries”, J. Operator Theory, Vol. 47, (2002), pp. 169–186, http://front.math.ucdavis.edu/funct-an/9712007. [14] R. Exel and D. Royer: ”The crossed product by a partial endomorphism” Manuscript at arXiv:math.OA/0410192 v1 6 Oct 2004, http://front.math.ucdavis.edu/math.OA/0410192. [15] R. Exel and A. Vershik: ”C*-algebras of irreversible dynamical systems”, To appear in Canadian J. Math., http://front.math.ucdavis.edu/math.OA/0203185. [16] B.K. Kwa´sniewski and A.V. Lebedev: ”Maximal ideal space of a commutative coefficient algebra”, preprint, http://front.math.ucdavis.edu/math.OA/0311416. [17] M. Laca and I. Raeburn: “A semigroup crossed product arising in number theory”, J. London Math. Soc., Vol. 59, (1999), pp. 330–344. [18] A.V. Lebedev and A. Odzijewicz: ”Extensions of C ∗ -algebras by partial isometries”, Matem. Sbornik, Vol. 195(7), (2004), pp. 37–70, http://front.math.ucdavis.edu/math.OA/0209049. [19] A.V. Lebedev: ”Topologically free partial actions and faithful representations of partial crossed products”, preprint, http://front.math.ucdavis.edu/math.OA/0305105. [20] K. McClanachan: ”K-theory for partial crossed products by discrete groups”, J. Funct. Analysis, Vol 130, (1995), pp. 77–117. [21] G.J. Murphy: C ∗ -algebras and operator theory, Academic Press, 1990. [22] G.J. Murphy: ”Crossed products of C ∗ -algebras by endomorphisms”, Integral Equations Oper. Theory, Vol. 24, (1996), pp. 298–319. [23] D.P. O’Donovan: ”Weighted shifts and covariance algebras”, Trans. Amer. Math. Soc., Vol. 208, (1975), pp. 1–25. [24] W.L. Paschke: ”The Crossed Product of a C ∗ -algebra by an Endomorphism”, Proc. Amer. math. Soc., Vol. 80, (1980), pp. 113–118. [25] G.K. Pedersen: London, 1979.

C ∗ -algebras and their automorphism groups, Academic Press,

[26] P.J. Stacey: ”Crossed products of C ∗ -algebras by ∗ -endomorphisms”, J. Austral. Math. Soc., Vol. 54, (1993), pp. 204–212. [27] R.F. Williams: ”Expanding attractors”’, IHES Publ. Math., Vol. 54, (1973), pp. 204– 212.

CEJM 3(4) 2005 766–793

K–theory from the point of view of C ∗–algebras and Fredholm representations Alexandr S. Mishchenko

∗

Departments of Mathematics and Mechanics, Moscow State University, Leninskije Gory, 119899, Moscow, Russia

Received 3 January 2005; accepted 1 July 2005 Abstract: These notes represent the subject of five lectures which were delivered as a minicourse during the VI conference in Krynica, Poland, “Geometry and Topology of Manifolds”, May, 2-8, 2004. c Central European Science Journals. All rights reserved.

Keywords: Vector bundle, K-theory, C*-algebras, Fredholm operators, Fredholm representations MSC (2000): 19L, 19K, 19J25, 55N15

1

Introduction

In the second half of the last century, research commenced and developed in what is now called ”non-commutative geometry”. As a matter of fact, this term concentrates on a circle of problems and tools which originally was based on the quite simple idea of re-formulating topological properties of spaces and continuous mappings in terms of appropriate algebras of continuous functions. This idea looks very old (it goes back to the theorem of I.M. Gelfand and M.A. Naimark (see, for example [1]) on the one-to-one correspondence between the category of compact topological spaces and the category of commutative unital C ∗ -algebras), and was developed by different authors both in the commutative and in the non-commutative cases. The first to clearly proclaim it as an action program was Alain Connes in his book ”non-commutative geometry” [2]. The idea, along with commutative C ∗ -algebras (which can be interpreted as algebras of continuous functions on the spaces of maximal ideals), to also consider non-commutative ∗ E-mail: [email protected]

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algebras as functions on a non-existing ”non-commutative” space was so fruitful that it allowed the joining together of a variety of methods and conceptions from different areas such as topology, differential geometry, functional analysis, representation theory, asymptotic methods in analysis and resulted in mutual enrichment by new properties and theorems. One of classical problems of smooth topology, which consists in the description of topological and homotopy properties of characteristic classes of smooth and piecewiselinear manifolds, has been almost completely and exclusively shaped by the different methods of functional analysis that were brought to bear on it. Vice versa, attempts to formulate and to solve classical topological problems have led to the enrichment of the methods of functional analysis. It is typical that solutions of particular problems of a new area lead to the discovery of new horizons in the development of mathematical methods and new properties of classical mathematical objects. The following notes should not be considered as a complete exposition of the subject of non-commutative geometry. The lectures were devoted to topics of interest to the author and are indicative of his point of view in the subject. Consequently, the contents of the lectures were distributed as follows: (1) Topological K–theory as a cohomology theory. Bott periodicity. Relation between the real, the complex and the quaternionic K–theories. (2) Elliptic operators as the homology K–theory, Atiyah homology K–theory as an ancestor of KK–theory. (3) C ∗ –algebras, Hilbert C ∗ -modules and Fredholm operators. Homotopical point of view. (4) Higher signature, C ∗ –signature of non-simply connected manifolds.

2

Some historical remarks on the formation of non-commutative geometry

2.1 From Poincare duality to the Hirzebruch formula. The Pontryagin characteristic classes, though not homotopy invariants, are nevertheless closely connected with the problem of the description of smooth structures of given homotopy type. Therefore, the problem of finding all homotopy invariant Pontryagin characteristic classes was a very actual one. However, in reality, another problem turned out to be more natural. It is clear that Pontryagin classes are invariants of smooth structures on a manifold. For the purpose of the classification of smooth structures, the most suitable objects are not the smooth structures but the so-called inner homology of manifolds or, using contemporary terms, bordisms of manifolds. Already L.S. Pontryagin [3] conjectured that inner homology could be described in terms of some algebraic expression of Pontryagin classes, the so-called Pontryagin numbers. He established that the Pontryagin numbers are at least invariants of inner homology [4, Theorem 3]. W. Browder and S.P. Novikov were the first to prove that only the Pon-

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tryagin number which coincides with the signature of an oriented manifold is homotopy invariant. This fact was established by means of surgery theory developed in [5], [6]. The formula that asserts the coincidence of the signature with a Pontryagin number is known now as the Hirzebruch formula [7], though its special case was obtained by V.A. Rokhlin [8] a year before. Investigations of the Poincare duality and the Hirzebruch formula have a long history, which is partly related to the development of non-commutative geometry. Here we shall describe only some aspects that were typical of the Moscow school of topology. The start of that history should be located in the famous manuscript of Poincare in 1895 [9], where Poincare duality was formulated. Although the complete statement and its full proof were presented much later, one can, without reservations, regard Poincare as the founder of the theory. After that was required the discovery of homology groups (E. Noether, 1925) and cohomology groups (J.W. Alexander, A.N. Kolmogorov, 1934). The most essential was, probably, the discovery of characteristic classes (E.L. Stiefel, Y. Whitney (1935); L. Pontryagin (1947); S.S. Chern (1948)). The Hirzebruch formula is an excellent example of the application of categorical method as a basic tool in algebraic and differential topology. Indeed, Poincare seems always to indicate when he had proved the coincidence of the Betti numbers of manifolds which are equidistant from the ends. But after the introduction of the notion of homology groups, Poincare duality began to be expressed a little differently: as the equality of the ranks of the corresponding homology groups. At that time it was not significant what type of homology groups were employed, whether with integer or with rational coefficients, since the rank of an integer homology group coincides with the dimension of the homology group over rational coefficients. But the notion of homology groups allowed to enrich ⇒ to expand Poincare duality by consideration of the homology groups over finite fields. Taking into account torsions of the homology groups, one obtained isomorphisms of some homology groups, but not in the same dimensions where the Betti numbers coincide. This apparent inconsistency was understood after the discovery of the cohomology groups and their duality to the homology groups. Thus finally, Poincare duality became sound as an isomorphism between the homology groups and the cohomology groups Hk (M; Z) = H n−k (M; Z).

(1)

The crucial understanding here is that the Poincare duality is not an abstract isomorphism of groups, but the isomorphism generated by a natural operation in the category of manifolds. For instance, in a special case of middle dimension for even-dimensional manifolds (dim M = n = 2m) with rational coefficients, the condition (1) becomes trivial since H m (M; Q) = Hom (Hm (M; Q), Q) ≡ Hm (M; Q).

(2)

But in the equation (2), the isomorphism between the homology groups and the cohomology groups is not chosen at will. Poincare duality says that there is the definite

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homomorphism generated by the intersection of the fundamental cycle [M] ∩[M ] : H n−k (M; Q)−→Hk (M; Q). This means that the manifold M gives rise to a non-degenerate quadratic form which has an additional invariant — the signature of the quadratic form. The signature plays a crucial role in many problems of differential topology.

2.2

Homotopy invariants of non-simply connected manifolds.

This collection of problems is devoted to finding the most complete system of invariants of smooth manifolds. In a natural way the smooth structure generates on a manifold a system of so-called characteristic classes, which take values in the cohomology groups with a different system of coefficients. Characteristic classes not only have natural descriptions and representations in differential geometric terms, but their properties also allow us to classify the structures of smooth manifolds in practically an exhaustive way modulo a finite number of possibilities. Consequently, the theory of characteristic classes is a most essential tool for the study of geometrical and topological properties of manifolds. However, the system of characteristic classes is in some sense an over-determined system of data. More precisely, this means that for some characteristic classes their dependence on the choice of smooth structure is inessential. Therefore, one of the problems was to find out to what extent one or other characteristic class is invariant with respect to an equivalence relation on manifolds. The best known topological equivalence relations between manifolds are piece-linear homeomorphisms, continuous homeomorphisms, homotopy equivalences and bordisms. For such kinds of relations one can formulate a problem: which characteristic classes are: a) combinatorially invariant, b) topologically invariant, c) homotopy invariant. The last relation (bordism) gives a trivial description of the invariance of characteristic classes: only characteristic numbers are invariant with respect to bordisms. Let us now restrict our considerations to rational Pontryagin classes. S.P. Novikov has proved (1965) that all rational Pontryagin classes are topologically invariant. In the case of homotopy invariance, at the present time the problem is very far from being solved. On the other hand, the problem of homotopy invariance of characteristic classes seems to be quite important on account of the fact that the homotopy type of manifolds seems to be more accessible to classification in comparison with its topological type. Moreover, existing methods of classification of smooth structures on a manifold can reduce this problem to a description of its homotopy type and its homology invariants. Thus, the problem of homotopy invariance of characteristic classes seemed to be one of the essential problems in differential topology. In particular, the problem of homotopy invariance of rational Pontryagin classes happened to be the most interesting (and probably the most difficult) from the point of view of mutual relations. For example, the importance of the problem is confirmed by that fact that the classification of smooth structures on a manifold by means the Morse surgeries demands a description of all

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homotopy invariant rational Pontryagin classes. In the case of simply connected manifolds, the problem was solved by Browder and Novikov who have proved that only signature is a homotopy invariant rational Pontryagin number. For non-simply connected manifolds, the problem of the description of all homotopy invariant rational Pontryagin classes which are responsible for obstructions to surgeries of normal mappings to homotopy equivalence, turned out to be more difficult. The difficulties are connected with the essential role that the structure of the fundamental group of the manifold plays here. This circumstance is as interesting as the fact that the description and identification of fundamental groups in finite terms is impossible. In some simple cases when the fundamental group is, for instance, free abelian, the problem could be solved directly in terms of differential geometric tools. In the general case, it turned out that the problem can be reduced to the one that the so-called higher signatures are homotopy invariant. The accurate formulation of this problem is known as the Novikov conjecture. A positive solution of the Novikov conjecture may permit, at least partly, the avoidance of algorithmic difficulties of description and the recognition of fundamental groups in the problem of the classification of smooth structures on non-simply connected manifolds. The Novikov conjecture says that any characteristic number of kind signx (M) = hL(M)f ∗ (x), [M]i is a homotopy invariant of the manifold M , where L(M) is the full Hirzebruch class, x ∈ H ∗ (Bπ; Q) is an arbitrary rational cohomology class of the classifying space of the fundamental group π = π1 (M) of the manifold M, f : M−→Bπ is the isomorphism of fundamental groups induced by the natural mapping. The numbers signx (M) are called higher signatures of the manifold M to indicate that when x = 1 the number sign1 (M) coincides with the classical signature of the manifold M. The situation with non-simply connected manifolds turns out to be quite different from the case of simply connected manifolds in spite of the fact that C. T. C. Wall had constructed a non-simply connected analogue of Morse surgeries. The obstructions to such kinds of surgeries does not have an effective description. One way to avoid this difficulty is to find out which rational characteristic classes for non-simply connected manifolds are homotopy invariant. Here we should define more accurately what we mean by characteristic classes for non-simply connected manifolds. As was mentioned above, we should consider only such invariants for a non-simply connected manifold as a) can be expressed in terms of the cohomology of the manifold and b) are invariants of nonsimply connected bordisms. In other words, each non-simply connected manifold M with fundamental group π = π1 (M) has a natural continuous map fM : M−→Bπ which induces an isomorphism of fundamental groups ≈

(fM )∗ : π = π1 (M)−→π1 (Bπ) = π.

(3)

Then the bordism of a non-simply connected manifold M is the singular bordism [M, fM ] ∈ Ω(Bπ) of the space Bπ. Hence, from the point of view of bordism theory, a rational characteristic number for the singular bordism [M, fM ] is a number of the following form ∗ ∗ α([M, fM ]) = hP (p1 (M), . . . , pn (M); fM (x1 ), . . . , fM (xk )) , [M]i ,

(4)

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where pj (M) are the Pontryagin classes of the manifold M, xj ∈ H ∗ (Bπ; Q) are arbitrary cohomology classes. Following the classical paper of R.Thom ([14]) one can obtain the result that the characteristic numbers of the type (4) form a complete system of invariants of the group Ω∗ (Bπ) ⊗ Q. Using the methods developed by C. T. C. Wall ([13]) one can prove that only higher signatures of the form ∗ sign x (M) = hL(M)fM (x); [M]i

(5)

may be homotopy invariant rational characteristic numbers for a non-simply connected manifold [M].

3

Topological K–theory

3.1 Locally trivial bundles, their structure groups, principal bundles Definition 3.1. Let E and B be two topological spaces with a continuous map p : E−→B. The map p is said to define a locally trivial bundle if there is a topological space F such that for any point x ∈ B there is a neighborhood U ∋ x for which the inverse image p−1 (U) is homeomorphic to the Cartesian product U × F . Moreover, it is required that the homeomorphism ϕ preserves fibers. This means in algebraic terms that the diagram ϕ

U × F −→ p−1 (U) ⊂ E    p  p y yπ y U

=

U

⊂B

is commutative where π : U × F −→U,

π(x, f ) = x

is the projection onto the first factor. The space E is called total space of the bundle or the fiberspace, the space B is called the base of the bundle, the space F is called the fiber of the bundle and the mapping p is called the projection. One can give an equivalent definition using the so-called transition functions: Definition 3.2. Let B and F be two topological spaces and {Uα } be a covering of the space B by a family of open sets. The system of homeomorphisms which form the commutative diagram ϕαβ

(Uα ∩ Uβ ) × F −→ (Uα ∩ Uβ ) × F     y y (Uα ∩ Uβ )

=

(Uα ∩ Uβ

(6)

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and satisfy the relations ϕαγ ϕγβ ϕβα = id, for any three indices α, β, γ on the intersection (Uα ∩ Uβ ∩ Uγ ) × F

(7)

ϕαα = id for each α. By analogy with the terminology for smooth manifolds, the open sets Uα are called charts, the family {Uα } is called the atlas of charts, the homeomorphisms ϕα

Uα × F −→ p−1 (Uα ) ⊂ E       y y y Uα

=

Uα

(8)

⊂B

are called the coordinate homeomorphisms and the ϕαβ are called the transition functions or the sewing functions. Two systems of the transition functions ϕβα , and ϕ′βα define isomorphic locally trivial bundles iff there exist fiber-preserving homeomorphisms h

α Uα × F −→ Uα × F     y y

Uα

=

Uα

such that ′ ϕβα = h−1 β ϕβα hα .

(9)

Let Homeo (F ) be the group of all homeomorphisms of the fiber F . Each fiberwise homeomorphism ϕ : U × F −→U × F, (10) defines a map ϕ : U−→Homeo (F ),

(11)

So instead of ϕαβ a family of functions ϕαβ : Uα ∩ Uβ −→Homeo (F ), can be defined on the intersection Uα ∩ Uβ and having values in the group Homeo (F ). The condition (7) means that ϕαα (x) = id, ϕαγ (x)ϕγβ (x)ϕβα (x) = id. x ∈ Uα ∩ Uβ ∩ Uγ .

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and we say that the cochain {ϕαβ } is a cocycle. The condition (9) means that there is a zero-dimensional cochain hα : Uα −→Homeo (F ) such that ′ ϕβα (x) = h−1 β (x)ϕβα (x)hα (x), x ∈ Uα ∩ Uβ .

Using the language of homological algebra the condition (9) means that cocycles {ϕβα } and {ϕ′βα } are cohomologous. Thus the family of locally trivial bundles with fiber F and base B is in one-to-one correspondence with the one-dimensional cohomology of the space B with coefficients in the sheaf of germs of continuous Homeo (F )–valued functions for the given open covering {Uα }. Despite obtaining a simple description of the family of locally trivial bundles in terms of homological algebra, it is ineffective since there is no simple method for calculating cohomologies of this kind. Nevertheless, this representation of the transition functions as a cocycle turns out to be very useful because of the situation described below. First of all, notice that using the new interpretation, a locally trivial bundle is determined by the base B, the atlas {Uα } and the functions {ϕαβ } taking values in the group G = Homeo (F ). The fiber F itself does not directly take part in the description of the bundle. Hence, one can at first describe a locally trivial bundle as a family of functions {ϕαβ } with values in some topological group G, and thereafter construct the total space of the bundle with fiber F by additionally defining an action of the group G on the space F , that is, defining a continuous homomorphism of the group G into the group Homeo (F ). Secondly, the notion of locally trivial bundle can be generalized and the structure of bundle made richer by requiring that both the transition functions ϕαβ and the functions hα not be arbitrary but take values in some subgroup of the homeomorphism group Homeo (F ). Thirdly, sometimes information about locally trivial bundle may be obtained by substituting some other fiber F ′ for the fiber F but using the ‘same’ transition functions. Thus, we come to a new definition of a locally trivial bundle with additional structure — the group where the transition functions take their values, the so-called the structure group. Definition 3.3. A locally trivial bundle with the structure group G is called a principal G–bundle if F = G and the action of the group G on F is defined by the left translations. Theorem 3.4. Let p : E−→B be a principal G–bundle. Then there is a right action of the group G on the total space E such that: 1) the right action of the group G is fiberwise, 2) the homeomorphism ϕ−1 α transforms the right action of the group G on the total space into right translations on the second factor. Using the transition functions it is very easy to define the inverse image of the bundle. Namely, let p : E−→B be a locally trivial bundle with structure group G and the collec-

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tion of transition functions ϕαβ : Uαβ −→G and let f : B ′ −→B be a continuous mapping. Then the inverse image f ∗ (p : E−→B) is defined as a collection of charts Uα′ = f −1 (Uα ) and a collection of transition functions ϕ′αβ (x) = ϕαβ (f (x)). Another geometric definition of the inverse bundle arises from the diagram E ′ = f ∗ (E) ⊂ E × B ′ −→ E       y y y B′

=

B′

(12)

f

−→ B

where E ′ consists of points (e, b′ ) ∈ E×B ′ such that f (b′ ) = p(e). The map fb : f ∗ (E)−→E is canonically defined by the map f . Theorem 3.5. Let ψ : E ′ −→E

(13)

be a continuous map of total spaces for principal G–bundles over bases B ′ and B. The map (13) is generated by a continuous map f : B ′ −→B if and only if the map ψ is equvivariant (with respect to right actions of the structure group G on the total spaces).

3.2 Homotopy properties, classifying spaces Theorem 3.6. The inverse images with respect to homotopic mappings are isomorphic bundles. Therefore, the category of all bundles with structure group G, BndlsG (B) forms a homotopy functor from the category of CW-spaces to the category of sets. Definition 3.7. A principal bundle p : EG −→BG is called a classifying bundle iff for any CW–space B there is a one-to-one correspondence BndlsG (B) ≈ [B, BG ]

(14)

generated by the map ϕ : [B, BG ]−→BndlsG (B), (15) ϕ(f ) = f ∗ (p : EG −→BG ). Theorem 3.8. The principal G–bundle, pG : EG −→BG

(16)

is a classifying bundle if all homotopy groups of the total space EG are trivial: πi (EG ) = 0, 0 ≤ i < ∞.

(17)

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3.3 Characteristic classes Definition 3.9. A mapping α : BndlsG (B)−→H ∗ (B) is called a characteristic class if the following diagram is commutative α

BndlsG (B) −→ H ∗ (B)    ∗  ∗ yf yf

(18)

α

BndlsG (B ′ ) −→ H ∗ (B ′ ) for any continuous mapping f : B ′ −→B, that is, α is a natural transformation of functors. Theorem 3.10. The family of all characteristic classes is in one-to-one correspondence with the cohomology H ∗ (BG ) by the assignment α(x)(ξ) = f ∗ (x) for ∗

x ∈ H (BG ),

f : B−→BG ,

(19) ∗

ξ = f (p : EG −→BG ).

3.4 Vector bundles, K–theory, Bott periodicity Members of the special (and very important) class of locally trivial bundles are called (real) vector bundles with structure groups GL(n, R) and fiber Rn . The structure group can be reduced to the subgroup O(n). If the structure group O(1) can be reduced to the subgroup G = SO(1) then the vector bundle is trivial and is denoted by 1. Similar versions arise for other structure groups: 1) Complex vector bundles with the structure group GL(n, C) and fiber Cn . 2) Quaternionic vector bundles with structure group GL(n, K) and fiber Kn , where K is the (non-commutative) field of quaternions. All of them admit useful algebraic operations: 1. Direct sum, ξ = ξ1 ⊕ ξ2 , 2. tensor product,ξ = ξ1 ⊗ ξ2 , 3. other tensor operations, HOM (ξ1 , ξ2),Λk (ξ). etc. Let ξ be a vector bundle over the field F = R, C, K. Let Γ(ξ) be the space of all sections ξ of the bundle. Then Γ(1) = C(B) — the ring of continuous functions with values in F . Theorem 3.11. The space Γ(ξ) has a natural structure of (left) C(B)–module by fiberwise multiplication. If B is a compact space, then Γ(ξ) is a finitely generated projective C(B)– module. Conversely, each finitely generated projective C(B)–module can be presented as a space Γ(ξ) for a vector bundle ξ. The property of compactness of B is essential for Γ(ξ) to be a finitely generated module.

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Definition 3.12. Let K(X) denotes the abelian group where the generators are (isomorphism classes of) vector bundles over the base X subject to the following relations: [ξ] + [η] − [ξ ⊕ η] = 0

(20)

for vector bundles ξ and η, and where [ξ] denotes the element of the group K(X) defined by the vector bundle ξ. The group defined in Definition 3.12 is called the Grothendieck group of the category of all vector bundles over the base X. To avoid confusion, the group generated by all real vector bundles will be denoted by KO (X), the group generated by all complex vector bundles will be denoted by KU (X) and the group generated by all quaternionic vector bundles will be denoted by KSp (X). Let K 0 (X, x0 ) denote the kernel of the homomorphism K(X)−→K(x0 ): K 0 (X, x0 ) = Ker (K(X)−→K(x0 )) . Elements of the subring K 0 (X, x0 ) are represented by differences [ξ]−[η] for which dim ξ = dim η. The elements of the ring K(X) are called virtual bundles and elements of the ring K 0 (X, x0 ) are virtual bundles of trivial dimension over the point x0 . Now consider a pair (X, Y ) of the cellular spaces, Y ⊂ X. Denote by K 0 (X, Y ) the ring K 0 (X, Y ) = K 0 (X/Y, [Y ]) = ker(K(X/Y )−→K([Y ])) where X/Y is the quotient space where the subspace Y is collapsed to a point [Y ]. For any negative integer −n, let K −n (X, Y ) = K 0 (S n X, S n Y ) where S n (X) denotes the n–times suspension of the space X: S n X = (S n × X)/(S n ∨ X). Theorem 3.13. The pair (X, Y ) induces an exact sequence K 0 (Y, x0 )←−K 0 (X, x0 )←−K 0 (X, Y )←− ←−K −1 (Y, x0 )←−K −1 (X, x0 )←−K −1 (X, Y )←−

←−K −2 (Y, x0 )←−K −2 (X, x0 )←−K −2 (X, Y )←− . . .

...................................................

. . . ←−K −n (Y, x0 )←−K −n (X, x0 )←−K −n (X, Y )←− . . .

...................................................

(21)

Consider a complex n–dimensional vector bundle ξ over the base X and let p : E−→X be the projection of the total space E onto the base X. Consider the space E, as a new base space, and a complex of vector bundles ϕ0

ϕ1

ϕ2

ϕn−1

0−→Λ0 η −→Λ1 η −→Λ2 η −→ . . . −→Λn η−→0,

(22)

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where η = p∗ ξ is the inverse image of the bundle ξ, Λk η is the k-skew power of the vector bundle η and the homomorphism ϕk : Λk η−→Λk+1 η is defined as exterior multiplication by the vector y ∈ E, y ∈ ξx , x = p(y). It is known that if the vector y ∈ ξx is non-zero, y 6= 0, then the complex (22) is exact. Consider the subspace D(ξ) ⊂ E consisting of all vectors y ∈ E such that |y| ≤ 1 with respect to a fixed Hermitian structure on the vector bundle ξ. Then the subspace S(ξ) ⊂ D(ξ) of all unit vectors gives the pair (Dξ), S(ξ)) for which the complex (22) is exact on S(ξ). Denote the element defined by (22) by β(ξ) ∈ K 0 (D(ξ), S(ξ)). Then, one has the homomorphism given by multiplication by the element β(ξ) β : K(X)−→K 0 (D(ξ), S(ξ)) .

(23)

The homomorphism (23) is an isomorphism called the Bott homomorphism. In particular the Bott element β ∈ K 0 (S2 , s0 ) = K −2 (S0 , s0 ) = Z generates a homomorphism e h: ⊗β K −n (X, Y )−→K −(n+2) (X, Y ) (24)

which is called the Bott periodicity isomorphism and hence forms a periodic cohomology K–theory.

3.5 Relations between complex, symplectic and real bundles Let G be a compact Lie group. A G–space X is a topological space X with continuous action of the group G on it. The map f : X−→Y is said to be equivariant if f (gx) = gf (x), g ∈ G. Similarly, if f is a locally trivial bundle and also equivariant then f is called an equivariant locally trivial bundle. An equivariant vector bundle is defined similarly. The theory of equivariant vector bundles is very similar to the classical theory. In particular, equivariant vector bundles admit the operations of direct sum and tensor product. In certain simple cases the description of equivariant vector bundles is reduced to the description of the usual vector bundles. The category of G– equivariant vector bundles is good place to give consistent descriptions of three different structures on vector bundles — complex, real and symplectic. Consider the group G = Z2 and a complex vector bundle ξ over the G–space X. This means that the group G acts on the space X. Let E be the total space of the bundle ξ and let p : E−→X be the projection in the definition of the vector bundle ξ. Let G act on the total space E as a fiberwise operator which is linear over the real numbers and anti-complex over complex numbers, that is, if τ ∈ G = Z2 is the generator then τ (λx) = λτ (x), λ ∈ C, x ∈ E.

(25)

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A vector bundle ξ with the action of the group G satisfying the condition (25) is called a KR–bundle . The operator τ is called the anti-complex involution. The corresponding Grothendieck group of KR–bundles is denoted by KR(X). Below we describe some of the relations with classical real and complex vector bundles. Proposition 3.14. Suppose that the G–space X has the form X = Y × Z2 and the involution τ transposes the second factor. Then the group KR(X) is naturally isomorphic to the group KU (Y ) and this isomorphism coincides with restriction of a vector bundle to the summand Y × {1}, 1 ∈ G = Z2 , ignoring the involution τ . Proposition 3.15. Suppose the involution τ on X is trivial. Then KR(X) ≈ KO (X).

(26)

The isomorphism (26) associates to any KR–bundle the fixed points of the involution τ . Proposition 3.16. The operation of forgetting the involution induces a homomorphism KR(X)−→KU (X) and when the involution is trivial on the base X this homomorphism coincides with complexification c : KO (X)−→KU (X). Moreover, the proof of Bott periodicity can be extended word by word to KR–theory: Theorem 3.17. There is an element β ∈ KR(D 1,1 , S 1,1) = KR−1,−1 (pt) such that the homomorphism given by multiplication by β β : KRp,q (X, Y )−→KRp−1,q−1 (X.Y )

(27)

is an isomorphism. It turns out that this scheme can be modified so that it includes another type of K–theory – that of quaternionic vector bundles. Let K be the (non-commutative) field of quaternions. As for real or complex vector bundles, we can consider locally trivial vector bundles with fiber K n and structure group GL(n, K), the so called quaternionic vector bundles. Each quaternionic vector bundle can be considered as a complex vector bundle p : E−→X with additional structure defined by a fiberwise anti-complex linear operator J such that J 2 = −1, IJ + JI = 0, where I is fiberwise multiplication by the imaginary unit. More generally, let J be a fiberwise anti-complex linear operator which acts on a complex vector bundles ξ and satisfies J 4 = 1, IJ + JI = 0.

(28)

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Then, the vector bundle ξ can be split into two summands ξ = ξ1 ⊕ ξ2 both invariant under the action of J, that is, J = J1 ⊕ J2 such that J12 = 1, J22 = −1.

(29)

Hence, the vector bundle ξ1 is the complexification of a real vector bundle and ξ2 is a quaternionic vector bundle. Consider a similar situation over a base X with an involution τ such that the operator (28) commutes with τ . Such a vector bundle will be called a KRS–bundle . Lemma 3.18. A KRS–bundle ξ is split into an equivariant direct sum ξ = ξ1 ⊕ ξ2 such that J 2 = 1 on ξ1 and J 2 = −1 on ξ2 . Lemma 3.18 shows that the Grothendieck group KRS(X) generated by KRS–bundles has a Z2 –grading, that is, KRS(X) = KRS0 (X) ⊕ KRS1 (X). It is clear that KRS0 (X) = KR(X). In the case when the involution τ acts trivially, KRS1 (X) = KQ (X), that is, KRS(X) = KO (X) ⊕ KQ (X) where KQ (X) is the group generated by quaternionic bundles.

q

KO

KQ

-8 Z α = γ2 Z uγ 2

-7

-6

-5

0

0

0

Z2

Z2

Z

h2 γ

hγ

γ

0

-4 Z uγ

-3

0

0

-2

-1

0

Z2

Z2

Z

h2

h

u2 = 4

0

0

Z u

Fig. 1 A list of the groups KO and KQ .

4

Elliptic operators as the homology K–theory, Atiyah homology K–theory as an ancestor of KK–theory

4.1 Homology K–theory. Algebraic categorical setting A naive point of view of homology theory is that the homology groups dual to the cohomology groups h∗ (X) should be considered as def

h∗ (X) = Hom (h∗ (X), Z).

(30)

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This naive definition is not good since it gives a non-exact functor. A more appropriate definition is the following. Consider a natural transformation of functors αY : h∗ (X × Y )−→h∗ (Y )

(31)

which is the homomorphism of h∗ (Y )–modules and for a continuous mapping f : Y ′ −→Y gives the commutative diagram α

h∗ (X × Y ) −→ h∗ (Y )     ∗ ∗ y(id × f ) yf

(32)

α

h∗ (X × Y ′ ) −→ h∗ (Y ′ ) Let be the family of all natural transformations of the type (31, 32). The functor h (X) defines a homology theory. ∗

4.2 PDO Consider a linear differential operator A which acts on the space of smooth functions of n real variables: A : C ∞ (Rn )−→C ∞ (Rn ). and is presented as a finite linear combination of partial derivatives A=

X

aα (x)

|α|≤m

∂ |α| . ∂xα

(33)

Put a(x, ξ) =

X

aα (x)ξ α i|α| .

|α|≤m

The function a(x, ξ) is called the symbol of a differential operator A. The operator A can be reconstructed from its symbol as   1 ∂ A = a x, . i ∂x Since the symbol is a polynomial with respect to the variables ξ, it can be split into homogeneous summands a(x, ξ) = am (x, ξ) + am−1 (x, ξ) + · · · + a0 (x, ξ). The highest term am (x, x) is called the principal symbol of the operator A while whole symbol is sometimes called the full symbol.

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Proposition 4.1. Let y = y(x) be a smooth change of variables. Then, in the new coordinate system the operator B defined by the formula (Bu)(y) = (Au (y(x)))x=x(y) is again a differential operator of order m for which the principal symbol is   ∂y(x(y)) bm (y, η) = am x(y), η . ∂x

(34)

The formula (34) shows that the variables ξ change as a tensor of valency (0, 1), that is, as components of a cotangent vector. The concept of a differential operator exists on an arbitrary smooth manifold M. The concept of a whole symbol is not well defined but the principal symbol can be defined as a function on the total space of the cotangent bundle T ∗ M. It is clear that the differential operator A does not depend on the principal symbol alone but only up to the addition of an operator of smaller order. The notion of a differential operator can be generalized in various directions. First of all, notice that (Au) (x) = Fξ−→x (a(x, ξ) (Fx−→ξ (u)(ξ))) , (35) where F is the Fourier transform. Hence, we can enlarge the family of symbols to include some functions which are not polynomials. Then the operator A defined by formula (35) with non-polynomial symbol is called a pseudodifferential operator of order m (more exactly, not greater than m). The pseudodifferential operator A acts on the Schwartz space S. This definition of a pseudodifferential operator can be extended to the Schwartz space of functions on an arbitrary compact manifold M. Let {Uα } be an atlas of charts with a local coordinate system xα . Let {ϕα } be a partition of unity subordinate to the atlas of charts, that is, X 0 ≤ ϕα (x) ≤ 1, ϕα (x) ≡ 1, supp ϕα ⊂ Uα . α

Let ψα (x) be functions such that

supp ψα ⊂ Uα , ϕα (x)ψα (x) ≡ ϕα (x). Define an operator A by the formula X A(u)(x) = ψα (x)Aα (ϕα (x)u(x)) ,

(36)

α

where Aα is a pseudodifferential operator on the chart Uα (which is diffeomorphic to Rn ) with principal symbol aα (xα , ξα ) = a(x, ξ). In general, the operator A depends on the choice of functions ψα , ϕα and the local coordinate system xα , uniquely up to the addition of a pseudodifferential operator of order strictly less than m.

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The next useful generalization consists of a change from functions on the manifold M to smooth sections of vector bundles. The crucial property of the definition (36) is the following Proposition 4.2. Let a : π ∗ (ξ1 )−→π ∗ (ξ2 ), b : π ∗ (ξa )−→π ∗ (ξ3 ) be two symbols of orders m1 , m2 . Let c = ba be the composition of the symbols. Then the operator b(D)a(D) − c(D) : Γ∞ (ξ1 )−→Γ∞ (ξ3 ) is a pseudodifferential operator of order m1 + m2 − 1. Proposition 4.2 leads to a way of solving equations of the form Au = f

(37)

for certain pseudodifferential operators A. To find a solution of (37), it suffices to construct a left inverse operator B, that is, BA = 1. Usually, this is not possible, but a weaker condition can be realized. Condition 4.3. a(x, ξ) is invertible for sufficiently large |ξ| ≥ C. The pseudodifferential operator A = a(D) is called an elliptic if Condition 4.3 holds. If A is elliptic operator than there is an (elliptic) operator B = b(D) such that AD − id is the operator of order -1. The final generalization for elliptic operators is the substitution of a sequence of pseudodifferential operators for a single elliptic operator. Let ξ1 , ξ2 , . . . , ξk be a sequence of vector bundles over the manifold M and let a

a

ak−1

1 2 ∗ 0−→π ∗ (ξ1 )−→π (ξ2 )−→ . . . −→π ∗ (ξk )−→0

(38)

be a sequence of symbols of order (m1 , . . . , mk−1 ). Suppose the sequence (38) forms a complex, that is, as as−1 = 0. Then the sequence of operators a1 (D)

0−→Γ∞ (ξ1 ) −→ Γ∞ (ξ2 )−→ . . . −→Γ∞ (ξk )−→0

(39)

in general, does not form a complex because we can only know that the composition ak (D)ak−1(D) is a pseudodifferential operator of the order less then ms + ms−1 . If the sequence of pseudodifferential operators forms a complex and the sequence of symbols (38) is exact away from a neighborhood of the zero section in T ∗ M then the sequence (39) is called an elliptic complex of pseudodifferential operators.

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4.3 Fredholm operators The bounded operator K : H−→H is said to be compact if any bounded subset X ⊂ H is mapped to a precompact set, that is, the set F (X) is compact. If dim Im (K) < ∞ then K is called a finite-dimensional operator. Each finite-dimensional operator is compact. If limn→∞ kKn − Kk = 0 and the Kn are compact operators, then K is again a compact operator. Moreover, each compact operator K can be presented as K = limn→∞ Kn , where the Kn are finite-dimensional operators. The operator F is said to be Fredholm if there is an operator G such that both K = F G − 1 and K ′ = GF − 1 are compact. Theorem 4.4. Let F be a Fredholm operator. Then (1) dim Ker F < ∞, dim Coker F < ∞ and the image, Im F , is closed. The number index F = dim Ker F − dim Coker F is called the index of the Fredholm operator F. (2) index F = dim Ker F − dim Ker F ∗ , where F ∗ is the adjoint operator. (3) there exists ε > 0 such that if kF − Gk < ε then G is a Fredholm operator and index F = index G, (4) if K is compact then F + K is also Fredholm and index (F + K) = index F.

(40)

(5) If F and G are Fredholm operators, then the composition F G is Fredholm and index (F G) = index F + index G. The notion of a Fredholm operator has an interpretation in terms of the finitedimensional homology groups of a complex of Hilbert spaces. In general, consider a sequence of Hilbert spaces and bounded operators d

d

dn−1

0 1 0−→C0 −→C 1 −→ . . . −→Cn −→0.

(41)

We say that the sequence (41) is a Fredholm complex if dk dk−1 0, Im dk is a closed subspace and dim (Ker dk /Coker dk−1) = dim H (Ck , dk ) < ∞. Then the index of Fredholm complex (41) is defined by the following formula: X index (C, d) = (−1)k dim H(Ck , dk ). k

Theorem 4.5. Let d

d

dn−1

0 1 0−→C0 −→C 1 −→ . . . −→Cn −→0

(42)

be a sequence satisfying the condition that each dk dk−1 is compact. Then the following conditions are equivalent:

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(1) There exist operators fk : Ck −→Ck−1 such that fk+1 dk + dk−1fk = 1 + rk where each rk is compact. (2) There exist compact operators sk such that the sequence of operators d′k = dk + sk forms a Fredholm complex. The index of this Fredholm complex is independent of the operators sk .

4.4 Sobolev spaces Consider an arbitrary compact manifold M and a vector bundle ξ. One can define a Sobolev norm on the space of the sections Γ∞ (M, ξ), using the formula kuk2s

=

Z

u(x)(1 + ∆)s u(x)dx,

Rn

where ∆ is the Laplace-Beltrami operator on the manifold M with respect to a Riemannian metric. The Sobolev norm depends on the choice of Riemannian metric, inclusion of the bundle ξ in the trivial bundle uniquely equivalent norms. Hence, the completion of the space of sections Γ∞ (M, ξ) is defined correctly. We shall denote this completion by Hs (M, ξ). Theorem 4.6. Let M be a compact manifold, ξ be a vector bundle over M and s1 < s2 . Then the natural inclusion Hs2 (M, ξ)−→Hs1 (M, ξ)

(43)

a(D) : Γ∞ (M, ξ1 )−→Γ∞ (M, ξ2 )

(44)

is a compact operator. Theorem 4.7. Let

be a pseudodifferential operator of order m. Then there is a constant C such that ka(D)uks−m ≤ Ckuks ,

(45)

that is, the operator a(D) can be extended to a bounded operator on Sobolev spaces: a(D) : Hs (M, ξ1 )−→Hs−m (M, ξ2 ).

(46)

Using theorems 4.6 and 4.7 it can be shown that an elliptic operator is Fredholm for appropriate choices of Sobolev spaces. Theorem 4.8. Let a(D) be an elliptic pseudodifferential operator of order m as in (44). Then its extension (46) is Fredholm. The index of the operator (46) is independent of the choice of the number s.

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4.5 Index of elliptic operators An elliptic operator σ(D) is defined by a symbol σ : π ∗ (ξ1 )−→π ∗ (ξ2 )

(47)

which is an isomorphism away from a neighborhood of the zero section of the cotangent bundle T ∗ M. Since M is a compact manifold, the symbol (47) defines a triple (π ∗ (ξ1 ), σ, π ∗ (ξ2 )) which in turn defines an element [σ] ∈ K(D(T ∗ M), S(T ∗ M)) = Kc (T ∗ M), where Kc (T ∗ M) denotes the K–groups with compact supports. Theorem 4.9. The index index σ(D) of the Fredholm operator σ(D) depends only on the element [σ] ∈ Kc (T ∗ M). The mapping index : Kc (T ∗ M)−→Z is an additive homomorphism. In addition, index σ(D) = p∗ [σ],

(48)

where p∗ : Kc (T ∗ M)−→Kc (pt) = Z is the direct image homomorphism induced by the trivial mapping p : M−→pt.

4.6 The Atiyah homology theory The naive idea is that cohomology K–group (with compact supports) of the total space of cotangent bundle of the manifold M, Kc (T ∗ M), should be isomorphic to a homology K–group due to a Poincare duality, K∗ (T ∗ M) ≈ K∗ (M). This identification can be arranged to be a natural transformation of functors D : Kc (T ∗ M) ≈ K∗ (M), D(σ) : K ∗ (M × N) −→ K ∗ (N)

(49)

D(σ)(ξ) = index (σ ⊗ ξ) ∈ K ∗ (N). Therefore, the homology K–groups, K∗ (M) can be identified with the collection of triples σ = (H, F, ρ), where H is a Hilbert space, F is a Fredholm operator, ρ : C(M)−→B(H) is a representation of the algebra C(M) of all continuous functions to the algebra B(H) of bounded operators, such that for any function f ∈ C(M) the operator F f − f F : H−→H is compact. If ξ is a vector bundle on M then the

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space Γ(ξ) is a finitely generated projective module over C(M). Therefore the operator F ⊗ idξ : H ⊗C(M ) Γ(ξ)−→H ⊗C(M ) Γ(ξ) is Fredholm. Hence, one obtains a natural transformation σ : K ∗ (M × N) −→ K ∗ (N),

(50)

σ(ξ) = index (F ⊗ idξ ) ∈ K ∗ (N). This definition was an ancestor of KK–theory.

5

C ∗–algebras, Hilbert C ∗–modules and C ∗–Fredholm operators

5.1 Hilbert C ∗–modules The simplest case of C ∗ –algebras is the case of commutative C ∗ –algebras. The Gelfand– Naimark theorem ([1]) says that any commutative C ∗ –algebra with unit is isomorphic to an algebra C(X) of continuous functions on a compact space X. This crucial observation leads to a simple but very useful definition of Hilbert modules over the C ∗ –algebra A. Following Paschke ([21]), the Hilbert A–module M is a Banach A–module with an additional structure of inner product hx, yi ∈ A, x, y ∈ M which possesses the natural properties of inner products. If ξ is a finite-dimensional vector bundle over a compact space X, then Γ(ξ) is a finitely generated projective Hilbert C(X)–module. And conversely, each finitely generated projective Hilbert module P over the algebra C(X) is isomorphic to a section module Γ(ξ) for some finite-dimensional vector bundle ξ. Therefore K0 (C(X)) ∼ = K(X).

5.2 Fredholm operators, Calkin algebra A finite-dimensional vector bundle ξ over a compact space X, can be described as a continuous family of projectors, that is a continuous matrix-valued function P = P (x), x ∈ X, P (x) ∈ Mat(N, N), P (x)P (x) = P (x), P (x) : CN −→CN . This means that ξ = Im P . Here Mat(N, N) denotes the space of N × N matrices. Hence if η = ker P then ξ ⊕ η = N. (51) P∞ P∞ Then ξ⊕ k=1 Nk ≈ k=1 Nk ≈ H ×X where H is a Hilbert space, or ξ⊕H ⊗X = H ×X. Hence, there is a continuous Fredholm family F (x) : H−→H, Ker (F ) = ξ, Coker (F ) = 0.

(52)

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And conversely, if we have a continuous family of Fredholm operators F (x) : H−→H such that dim Ker F (x) = const , dim Coker F (x) = const then both ξ = Ker F and η = Coker F are locally trivial vector bundles. More generally, for an arbitrary continuous family of Fredholm operators F (x) : H−→H there is a continuous compact family K(x) : H−→H such that the new family Fe(x) = F (x) + K(x) satisfies the conditions dim Ker Fe(x) = const , dim Coker Fe(x) = const consequently defining two vector bundles ξ and η which generate an element [ξ] − [η] ∈ K(X) not depending on the choice of compact family. This correspondence is actually one-to-one. In fact, if two vector bundles ξ and η are isomorphic then there is a compact deformation of F (x) such that Ker F (x) = 0, Coker F (x) = 0, that is F (x) is an isomorphism, F (x) ∈ G(H). The remarkable fact discovered by Kuiper ([22]) is that the group G(H) as a topological space is contractible, i.e. the space F (H) of Fredholm operators is a representative of the classifying space BU for vector bundles. In other words, one can consider the Hilbert space H and the group of invertible operators GL(H) ⊂ B(H). The Kuiper theorem says that πi (GL(H)) = 0, 0 ≤ i < ∞. (53)

5.3 K–theory for C ∗–algebras, Chern character Generalization of K–theory for C ∗ –algebra A. KA (X) is the Grothendieck group generated by vector bundles whose fibers M are finitely generated projective A–modules, and the structure groups Aut A (M). KA∗ (X) are the corresponding periodic cohomology theory. For example, let us consider the quotient algebra Q(H) = B(H)/K(H), the so-called Calkin algebra, where B(H) is the algebra of bounded operators of the Hilbert space H, K(H) is the algebra of compact operators. Let p : B(H)−→Q(H) be the natural projector. Then the Fredholm family F (x) : H−→H generates the family F : X−→Q(H), F (x) = p(F (x)), F (x) is invertible that is F : X−→G(Q(H)). So, one can prove that the space G(Q(H)) represents the classifying space BU for vector bundles. In other words, 1 K 0 (X) ∼ (X). = KQ(H) A generalization of the Kuiper theorem for the group GL∗A (l2 (A)) of all invertible operators which admit adjoint operators. Let F be the space of all Fredholm operators. Then K ∗ (X) ≈ [X, F ]. (54) Let Q = B(H)/K be the Calkin algebra, where K is the subalgebra of all compact operators. Let G(Q) be the group of invertible elements in the algebra Q. Then one has a homomorphism [X, F ]−→[X, Q],

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hence a homomorphism 1 K 0 (X)−→KQ (X)

which is an isomorphism. The Chern character chA : KA∗ (X)−→H ∗ (X; KA∗ (pt) ⊗ Q) is defined in a tautological way: let us consider the natural pairing K ∗ (X) ⊗ KA∗ (pt)−→KA∗ (X)

(55)

which generates the isomorphism θ

K ∗ (X) ⊗ KA∗ (pt) ⊗ Q−→KA∗ (X) ⊗ Q

(56)

due to the classical uniqueness theorem in axiomatic homology theory. Then, the Chern character is defined as the composition θ −1 ch chA : KA∗ (X) ⊂ KA∗ (X) ⊗ Q−→K ∗ (X) ⊗ (KA∗ (pt) ⊗ Q) −→

(57) ch −→H ∗ (X; KA∗ (pt) ⊗ Q). Therefore, the next theorem is also tautological: Theorem 5.1. If X is a finite CW –space, the Chern character induces the isomorphism chA : KA∗ (X) ⊗ Q−→H ∗ (X; KA∗ (pt) ⊗ Q).

5.4 Non-simply connected manifolds and canonical line vector bundle Let π be a finitely presented group which can serve as a fundamental group of a compact connected manifold M, π = π1 (M, x0 ). Let Bπ be the classifying space for the group π. Then there is a continuous mapping fM : M−→Bπ

(58)

such that the induced homomorphism (fM )∗ : π1 (M, x0 )−→π1 (Bπ, b0 ) = π

(59)

is an isomorphism. One can then construct the line vector bundle ξA over M with fiber A, a one-dimensional free module over the group C ∗ –algebra A = C ∗ [π] using the representation π ⊂ C∗ [π]. This canonical line vector bundle can be used to construct the so-called assembly map µ : K∗ (Bπ)−→K∗ (C∗ [π]) (60)

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5.5 Symmetric equivariant C ∗–signature Let M be a closed oriented non-simply connected manifold with fundamental group π. Let Bπ be the classifying space for the group π and let fM : M−→Bπ, be a map inducing the isomorphism of fundamental groups. Let Ω∗ (Bπ) denote the bordism group of pairs (M, fM ). Recall that Ω∗ (Bπ) is a module over the ring Ω∗ = Ω∗ ( pt ). One can construct a homomorphism σ : Ω∗ (Bπ)−→L∗ (Cπ)

(61)

which for every manifold (M, fM ) assigns the element σ(M) ∈ L∗ (Cπ), the so-called symmetric Cπ–signature, where L∗ (Cπ) is the Wall group for the group ring Cπ. The homomorphism σ satisfies the following conditions: (a) σ is homotopy invariant, (b) if N is a simply connected manifold and τ (N) is its signature then σ(M × N) = σ(M)τ (N) ∈ L∗ (Cπ). We shall be interested only in the groups after tensor multiplication with the field Q, in other words, in the homomorphism σ : Ω∗ (Bπ) ⊗ Q−→L∗ (Cπ) ⊗ Q. However, Ω∗ (Bπ) ⊗ Q ≈ H∗ (Bπ; Q) ⊗ Ω∗ . Hence one has σ : H∗ (Bπ; Q)−→L∗ (Cπ) ⊗ Q. Thus, the homomorphism σ represents the cohomology class σ ∈ H ∗ (Bπ; L∗ (Cπ) ⊗ Q). Then, for any manifold (M, fM ) one has ∗ σ(M, fM ) = hL(M)fM (σ), [M]i ∈ L∗ (Cπ) ⊗ Q.

(62)

Hence, if α : L∗ (Cπ) ⊗ Q−→Q is an additive functional and α(σ) = x ∈ H ∗ (Bπ; Q) then ∗ signx (M, fM ) = hL(M)fM (x), [M]i ∈ Q

should be the homotopy-invariant higher signature. This gives a description of the family of all homotopy-invariant higher signatures. Hence, one should study the cohomology class σ ∈ H ∗ (Bπ; L∗ (Cπ) ⊗ Q) = H ∗ (Bπ; Q) ⊗ L∗ (Cπ) ⊗ Q and look for all elements of the form α(σ) = x ∈ H ∗ (Bπ; Q).

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5.5.1 Combinatorial description of symmetric Cπ–signature Here we give an economical description of algebraic Poincare complexes as a graded free Cπ–module with the boundary operator and the Poincare duality operator. Consider a chain complex of Cπ–modules C, d: n M

C= d=

Ck ,

k=0 n M

dk ,

k=1

dk : Ck −→Ck−1 and a Poincare duality homomorphism D : C ∗ −→C,

deg D = n.

They form the diagram d

d

d

d∗

d∗n−1

d∗

n 1 2 C0 ←− C1 ←− · · · ←− Cn x x x    D0 D1 Dn

1 n ∗ C0∗ Cn∗ ←− Cn−1 ←− · · · ←−

with the following properties:

dk−1 dk = 0, dk Dk + (−1)

k+1

Dk−1 d∗n−k+1 = 0,

∗ Dk = (−1)k(n−k) Dn−k .

(63)

Assume that the Poincare duality homomorphism induces an isomorphism of homology groups. Then the triple (C, d, D) is called a algebraic Poincare complex. This definition permits the construction of the algebraic Poincare complex σ(X) for each triangulation of the combinatorial manifold X: σ(X) = (C, d, D), where C = C∗ (X; Cπ) is the graded chain complex of the manifold X with local system of coefficients induced by the natural inclusion of the fundamental group π = π1 (X) in the group ring Cπ, d is the boundary homomorphism, D = ⊕Dk ,

Dk =


1 ∩[X] + (−1)k(n−k) (∩[X])∗ , 2

where ∩[X] is the intersection with the open fundamental cycle of the manifold X. Put Fk = ik(k−1) Dk . (64)

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Then the diagram d

d

d

n 1 2 C0 ←− C1 ←− · · · ←− Cn x x x    F0 F1 Fn

d∗

d∗n−1

(65)

d∗

n 1 ∗ Cn∗ ←− Cn−1 ←− · · · ←− C0∗

possesses more natural conditions of commutativity and conjugacy dk Fk + Fk−1 d∗n−k+1 = 0, Fk = (−1)

n(n−1) 2

∗ Fn−k .

(66)

Let F = ⊕nk=0 Fk , F,

deg F = n.

Over completion to a regular C ∗ –algebra C ∗ [π], one can define an element of hermitian K–theory using the non-degenerate self-adjoint operator G = d + d∗ + F : C−→C. Then sign [C, G] = sign (C, d, D) ∈ K0h (C ∗ [π]).

6

(67)

Additional historical remarks

The only candidates which are homotopy invariant characteristic numbers are the higher signatures. Moreover, any homotopy invariant higher signature can be expressed from a universal symmetric equivariant signature of the non-simply connected manifold. Therefore, to look for homotopy invariant higher signatures, one can search through different geometric homomorphisms α : L∗ (Cπ) ⊗ Q−→Q. On of them is the so-called Fredholm representation of the fundamental group. Application of the representation theory in the finite-dimensional case leads to Hirzebruchtype formulas for signatures with local system of coefficients. But the collection of characteristic numbers which can be represented by means of finite-dimensional representations is not very large and in many cases reduces to the classical signature. The most significant here is the contribution by Lusztig ([28]) where the class of representations with indefinite metric is considered. The crucial step was to find a class of infinite-dimensional representations which preserve natural properties of the finite-dimensional representations. This infinite-dimensional analogue consists of a new functional-analytic construction as a pair of unitary infinite dimensional representations (T1 , T2 ) of the fundamental group π in the Hilbert space H and a Fredholm operator F which braids the representations T1 and T2 upto compact operators. The triple ρ = (T1 , F, T2 ) is called the Fredholm representation of the group π.

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From the categorical point of view, the Fredholm representation is a relative representation of the group C ∗ –algebra C ∗ [π] in the pair of Banach algebras (B(H), Q(H) where B(H) is the algebra of bounded operators on the Hilbert space H and Q(H) is the Calkin algebra Q(H) = B(H)/K(H). Then, for different classes of manifolds one can construct a sufficiently rich resource of Fredholm representations. For one class of examples from amongst many others, there are the manifolds with Riemannian metric of nonpositive sectional curvature, the so-called hyperbolic fundamental groups. For the most complete description of the state of these problems, one may consult ([29]) and the book ([2]).

References [1] M.A. Naimark: Normed algebras, Nauka, Moscow, 1968 (in Russian), English translation: Wolters–NoordHoff, 1972. [2] A. Connes: Noncommutative Geometry, Academic Press, 1994. [3] L.S. Pontryagin: “Classification of some fiber bundles”, Dokl. Akad. Nauk SSSR, Vol. 47(5), (1945), pp. 327–330 (in Russian). [4] L.S. Pontryagin: “Characteristic cycles of differntiable manifolds”, Mathem. Sbornik, Vol. 21(2), (1947), pp. 233–284 (in Russian). [5] W. Browder: Homotopy type of differential manifolds, Colloquium on Algebraic Topology, Aarhus Universitet, Aarhus, Matematisk Institut, 1962, pp. 42–46. [6] S.P. Novikov: “Homotopy equivalent smoth manifolds, I”, Izv.Akad.Nauk SSSR, Ser.Mat., Vol. 28, (1964), pp, 365–474. [7] F. Hirzebruch: “On Steenrod’s reduced powers, the index of interia, and the Todd genus”, Proc.Nat,Acad.Sci. U.S.A., Vol. 39, (1953), pp. 951–956. [8] V.A. Rokhlin: “New results in the theory of 4-dimensional manifolds”, Dokl. Akad. Nauk SSSR, Vol. 84, (1952), pp. 221–224. [9] A. Poincare: “Analysis situs”, Journal de l’Ecole Polytechniques, Vol. 1, (1895), pp. 1–121. [10] A.S. Mishchenko: “Homotopy invariant of non simply connected manifolds. Rational invariants. I”, Izv. Akad. Nauk. SSSR, Vol. 34(3), (1970), pp. 501–514. [11] A.S. Mishchenko: “Controlled Fredholm representations”, In: S.C. Ferry, A. Ranicki and J. Rosenberg (Eds.): Novikov Conjectures, Index Theorems and Rigidity, Vol. 1, London Math. Soc. Lect. Notes Series, Vol. 226, Cambridge Univ. Press, 1995, pp. 174–200. [12] A.S. Mishchenko: “C ∗ –algebras and K-theory”, Algebraic topology, Lecture Notes in Mathematics, Vol. 763, Aarhus, 1979, pp. 262–274. [13] C.T.C. Wall: Surgery of compact manifolds, Academic Press, New York, 1971. [14] R. Thom: “Quelques properietes globales des varietes differentiables”, Comm. Math. Helv., Vol. 28, (1954), pp. 17–86. [15] G. Luke and A.S. Mishchenko: Vector Bundles and Their Applications, Kluwer Academic Publishers, Dordrrecht, Boston, London, 1998. [16] D. Husemoller: Fiber Bundles, McGraw–Hill, N.Y., 1966.

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