Bq,
400
Chapter VII.
Tensor products of topological *-algebras
for all (p, q) G FA x Fg. Moreover, the defining families of seminorms for e, a, T and u> are all identical and this clearly gives (31.35). Note that Corollary 31.16 remains true for any *-admissible topology r on A (g> B with a -< r -< to. Questions of nuclearity for locally C*-algebras, in terms of injective, respectively projective, C*-convex tensor products, have been considered by S.J Bhatt and D.J. Karia in [57]. Let now X be a completely regular space, A[r^] a C*-convex algebra and C(X, A) the algebra of all A-valued continuous functions on X. For a given / e C(X, A) an involution is defined on C(X, A) by f*(x) := (/(x))*, for every x € X. Equip C(X, A) with the topology "c" of compact convergence; that is, if K stands for the family of all compact subsets of X, the topology c is given by the C*-seminorms QPMf)
sup{p(/(x)) : K G AC}, V / G C(X,A)
and (p,K) G T x K.
So CC(X,A) is a C*-convex algebra. When A[rr] is a locally C*-algebra and X a completely regular fc-space (take, for example, X to be locally compact), then CC(X, A) becomes complete, hence a locally C*-algebra (cf., for instance, [230, p. 231, Theorem 12]). 31.17 Theorem. Let A[TA] be a C*-convex algebra and B[TB] a unital commutative locally C* -algebra with locally equicontinuous spectrum (take, for example, B = CC(K)). Then, the following statements hold: (1) A <8> B e—> Cc(9Jl{B), A), up to a topological injective *-morphism. (2) When A[TA] is also complete (that is, a locally C*-algebra), one has A®B
= CC(9R(B),A),
up to a topological
*-isomorphism.
Proof. That e = a on A® B, follows from the proof of Corollary 31.16. (1) For z = YA=I Xi®Vi G A®B
and
V G Tt(B), define n
6:A®B^
Cc{Wl{B), A) : 9{z)(x>) := V
The map 0 is a *-homomorphism. Since B[TB] is *-semisimple (see Proposition 20.1) and commutative, n{ker(
31.
Tensor products of locally C*—algebras
401
topologically irreducible ^-representation of B[TQ) is 1-dimensional [276, Corollary (6.4)], and each character of B[TB] is *-preserving from Proposition 21.6(2) together with (21.6)). Hence, 6 is 1 — 1. To prove continuity of 0 and that of its inverse on the range, observe that local equicontinuity of 9Jl(B) implies continuity of the Gel'fand map of B[TB] as it follows from [262, p. 184, Corollary 1.3]. Thus (cf. Corollary 9.4), the following equality (31.36)
B[TB] = Cc(m(B)),
holds up to a topological *-isomorphism given by the Gel'fand map of B[TB],
such that (cf. proof of Theorem 9.3(4)) q{y) = \\y\\Kq,
(31.37)
V y e B
a n d q e T B.
This is due to the fact that the topology c in (31.36) is determined by the compacts Kq, q G FB, of $Jl(B) (see Lemma 9.1), given by Kq — Tl{B) n U°(e), 0 < £ < 1, for every q € rB.
Thus, using (31.37), as well
as the expression of p(x), x € A (respectively of q(y), y £ B) given after (29.7), we obtain QP,Kq(O(z))
= sup \p( ^2
:
= 8up(x'(Yl
:x'eU;(l),
= Bup{(p(jTlx'(xi)y?J :x'eU;(l), (Xi)yi) I Vi=1
J
= supL(Ylx'{xi)y^ = sup{ JTx'ix^y'iyi)
Kq
:x'eU;(l)\ )
:x'eU;(l)\ :x' eU°p{l), y' G t/g°(l)|
n
= £P,q(z), V z = ^2 xi <8> yi G A ® B and (p, q) € rA x rB. i=l
402
Chapter VII.
Tensor products of topological *-algebras
This completes the proof of (1). (2) Suppose now that A[T/I] is complete and note that 9Jl(B) being locally equicontinuous, is locally compact, therefore the C*-convex algebra Cc(Wl{B),A) is complete (cf. [230, p. 231, Theorem 2]), hence a locally C*-algebra. In particular,
Cc(m(B),A)=Cc(m(B))®A, with respect to a topological ^-isomorphism [262, p. 391, Theorem 1.1]. The assertion now follows from (31.36). Note that: (i) In the preceding Theorem 31.17, e = a = u> on A®B, according to Corollary 31.16. (ii) If -B[TJB] is a commutative Frechet topological algebra, local equicontinuity and local compactness are equivalent properties for its spectrum 9Jl(B) [262, p. 143]. 31.(2) The spectrum of an elementary tensor In this Subsection we present some estimations of the spectrum of an either arbitrary element or an elementary tensor in certain tensor product topological (*-) algebras by applying inverse limit preserving tensorial topologies. We first introduce some new notation. Let £J[r£;], .F^TT?] be locally convex spaces and r a compatible topology on E®F (Definition 29.3). Let (x',yr) € E' x F'. Then, the following continuous linear maps are defined (see [262, p. 441, Lemma 4.1]) xi
(resp. dy:) : E®F —> E (resp. F) n
:z = ^Xi®yi f resp.
n
i— 0x,(z)
(z) := Yl v\Vi)xi )
In particular, (31.39)
:=Y^x'(xi)yi
x'odyl =x'®y' = y'o-dx,.
31.
Tensor products of locally C* -algebras
403
31.18 Lemma. (1) Let A[TA] be an m-convex algebra with DJl(A) ^ 0 and B\TB\
an Arens-Michael algebra. Let r be an admissible topology on A®B
and if £ Wl(A).
Then d^ (cf. (31.38)) extends uniquely to a continuous
morphism from A®B in B[TB], also denoted by dv. T
(2) Let A[TA\ be a commutative symmetric advertibly complete m*convex algebra and B[TB] an Arens-Michael *-algebra.
Let r be a *-
admissible topology on A®B and if G Wl(A). Then, fl^ extends uniquely to a continuous *-morphism from A®B in B[TB\, also denoted by 'dip. T
Proof. The proof of (1) is included in (2). So we prove (2). By the properties of A[TA], y? is a ^-preserving morphism (cf. Proposition 21.6); namely, f is a continuous morphism with the property f{x*) = B in B\TB\-
Hence,
T
it is uniquely extended to a continuous linear map from A®B in B[TB\T
Using continuity of the multiplication and involution in A®B, it is proved T
that dtp is a *-morphism on A®B. The next Lemma 31.19 has been proved by A. Bonic for Banach algebras and the projective tensorial topology [69, Theorem 3.1]. Later on K.B. Laursen showed the same result for any admissible topology r [244, Lemma III.4]. 31.19 Lemma. Let A\TA\ be a unital commutative m-convex algebra and B[TB\)
a unital Arens-Michael algebra. Let r be an inverse limit preserv-
ing topology on A®B and z G A®B.
Then, z is left (respectively right)
T
invertible in A®B if and only ifd^z)
is left (respectively right) invertible
T
in B, for each if G Wl(A).
Proof. Suppose that z' is a left-inverse of z in A®B. Then, z'z = e^CSe^ T
and (see Lemma 31.18(1)) 'dv(z')idv>(z) — ^ ( e A ^ e e ) —
404
Chapter VII.
Tensor products of topological *-algebras
Conversely: Suppose that z is an element in A®B such that r
v{z)
(31.40)
is left-invertible in B, V f e Tl(A),
but z is not left-invertible in A®B. T
The idea of the proof, of this part of the lemma, is the following: We use the claim for z in (31.40) to construct a j ? e dJl(A) for which applying the first claim of (31.40) we are led to a contradiction. From Theorem 31.9, we have the following equality, up to a topological isomorphism (31.41)
A%B = limA p ®B q , (p,q) £ rA x rB. pq
T
Hence, since z is not left-invertible in A®B, there is an index (p,q) G T
A X -TB such that the component zpq of 2; is not left-invertible in the Banach algebra Ap®Bq. Thus, defining pq
Ipq := \ z'zpq : z' £ Ap®Bq \, pq
I
i
we get a proper left ideal in Ap®Bq. Then, there is a maximal left ideal pq
Mpq in Ap®Bq such that Ipg C Mpq. In particular, Mpq is closed [121, pq
p. 314, (B.5.2) Proposition]. Thus, considering the Banach space E := {Ap®Bq)/Mpq, with the natural quotient norm, we let [ipq be the left regular pq
^
representation of Ap®Bq on E; that is, pq
Hpq(u)(v
+ Mpq) : = uv + Mpq,
Vii.ue
Ap®Bq. pq
Since Mpq is a maximal left ideal, [ipq is algebraically irreducible [72, p. 120, Proposition 5]. On the other hand, it is easily seen that the linear operator fiPq(u) is bounded, for every u £ Ap®Bq. Let JC(E) be the Banach algebra of pq
all bounded linear operators on E. Then, fipq as an algebraically irreducible representation of the Banach algebra Ap®Bq on the Banach space E, such pq
that fJ,Pq(u) £ JC(E) (for all u 6 Ap®Bq),
is continuous (ibid., p. 128,
pq
Theorem 7). On the other hand, since jipq is algebraically irreducible,
31.
Tensor products of locally C* —algebras
405
we can apply Schur's lemma [72, p. 121, Proposition 6] to take that the commutant Im(fipq)c of the image of npq is a division (closed) subalgebra of C(E).
Hence (Gel'fand-Mazur theorem) it is topologically isomorphic
to C. Namely, Im(Mp(?)c = {\idE
(31.42)
:AeC}.
Further, in view of (31.41), fipq defines a representation /i of A®B on T
E, which is continuous since jipq has this property. That is, \x : A®B —> C(E) : u i—> fi(u) := npq(upq), T
where upq is the component of u in Ap®Bq.
An easy computation shows
pq
that Im(/u)c C Im(npq)c.
Therefore (see (31.42))
(31.43)
Im(M)c = {XidE : X G C}.
Moreover,
JJL{X
® eg) G Im(/i) c , for every x £ A.
Indeed: Let u £
Commutativity of A[TA] implies that fi(x <S> eg) commutes with
A®B. T
fi{u). Thus, taking into account continuity of /i, as well as continuity of the multiplication in £(E), we conclude that fj,(x <E> es) commutes with fj,(u), for all u G A®B. So, for every x G A, there is unique A T e C (see (31.43)) T
such that (31.44)
fi(x es) = KidE,
x G A.
This equality easily leads to an element ip of DJl(A) denned by
All properties of
corresponding properties of (i (in this regard, note that we may suppose q(es) — 1, for all q G FB, according to Theorem 2.3). Let now (x, y) G Ax B. Then, fi(x ®y) = /J,(xeA ® eBy) = [i(x ® eB)fi(eA ® y) = XxidE/J.(eA
<E> y) = M(e A O A x y) = ;u(eA ® ^ ( x y ) ) .
L i n e a r i t y a n d c o n t i n u i t y of \x give t h a t /x(u) = /j,(eA <S> ^ ( w ) ) , V U G A%B a n d <^ G 3Jt(A). T
406
Chapter VII.
Tensor products of topological *-algebras
Thus, if b is a left inverse of l&v>(z) in B with z as in (31.40) and ep, eq the identities in Ap and Bq respectively, we get (31.45)
fi(eA ® b)(fi(z)(ep ®eq + Mpq)) = ep ® eq + Mpq.
But the component zpq of z in Ap®Bq clearly belongs to Ipq, hence to PQ
Mpq; therefore zpq + Mpq = 0. So the definition of /x gives n(eA ® b)(n{z)(ep
eq + Mpq)) = n(eA b){zpq + Mpq) = 0
and this contradicts (31.45). Consequently (31.40) is false and so z is leftinvertible in B. The part of the lemma concerning right-invertibility is similarly proved. 31.20 Theorem. Let A\TA\ be a unital commutative m-convex algebra and B[TB] a unital Arens-Michael algebra. Let r be an inverse limit preserving topology on A® B and z G A®B. Then, the following hold: T
(1) z is invertible in A®B if and only if "dv(z) is invertible in B[TB], for
allipeWl(A). (2) spMB(z) = \J{spB(0v(z)) : y) - spA(x)spB{y), for all {x, y) G A x B. T
Proof. (1) It follows directly from Lemma 31.19. (2) Using (1) we deduce that A
i SPA®B(Z) O ^eA®eB-ze
G AgB
T
T
<£> «
ip{\eA
® e j j - z ) e G B , V v? G m(A)
AeB - ^ ( z ) 6 G B , V ^ e m(A)
« A i spuOM*)),
v
(3) It follows from (2) and (31.38), combined with (4.11) and (4.35) (also take into account the comments after (4.36)). In Subsection 34.(2), we apply essentially Theorem 31.20(2) to get hermiticity (respectively symmetry) of a tensor product Arens-Michael *algebra from the hermiticity (respectively symmetry) of the Arens-Michael s participating in the tensor product (Theorem 34.15).
31.
Tensor products of locally C* -algebras
407
The next result was first proved by A. Brown and C. Pearcy [88], in 1966, for C*-algebras of bounded linear operators on a Hilbert space. An abstract analogue of this result, was given by K.B. Laursen [244, p. 67, Corollary III.4], in 1969, in the context of unital symmetric Banach *algebras. 31.21 Corollary. Let A[TA\, B\TB\ be unital Arens-Michael*-algebras and T an inverse limit preserving topology on A® B. Let spA{x)spB{y)
= {AiA2 : Ai G spA{x),\2
€ spB(y)},
V (x,y) G A x B.
Then, for all (x, y) € A x B with either x G N(A) or y G N(B), (31.46)
spA§B(x
(g> y) C spA{x)sPB(y)
one has
U {0}.
T
When A[TA}} B\TQ\ are moreover hermitian, one has (31.47)
spMB(x
y) = spA{x)spB{y),
V (a;, y) G H(A) x H(B).
T
Proof. Let x G A with x G N(A), that is xx* = x*x. Then (see Theorem 4.13) there is a closed maximal commutative *-subalgebra M of A[T^], containing {eA,x},
such that
(31.48)
spM(z) = sPA{z), V
zeM.
Consider M ® B as a *-subalgebra of A.B and denote by C the closure T
of M <8> i? in A®B.
Recall that for a Banach algebra AQ and a closed
T
subalgebra Z?o of Ao, one has that spA0(xo) Q spDo{xo) U {0} , for every xQ € Do [327, p. 33, Theorem (1.6.12)]. Thus, in view of (31.21) we can apply Theorem 4.6(2) to get S
PA®B(X
® V) £ s P c ( s ® y) U {0}, V y G B.
r
But, from Theorem 31.20(3) and (31.48), we have (31.49)
spc(x (g> y) = spM(x)spB(y)
=
spA(x)spB{y),
408
Chapter VII.
Tensor products of topological *-algebras
for all x G N(A) and y £ B; this completes the proof of (31.46). Suppose now that Afr^], B\TB\ are moreover hermitian and let (x,y) G H(A) x H(B). Then, from Definition 21.1, we have spA{x) C 1 D spB{y), so that (see (31.46)) spA^B(x ® y) C R. Using again Theorem 4.6(2) T
(also cf. (31.21)) and the fact that for any unital Banach algebra AQ and xo £ Ao with spAo^o) Q K, one has that S£>A0(a;o) = spDo^o)) f° r a n y closed subalgebra Do of AQ containing XQ and the identity of AQ [239, p. 62; Corollary 2.3.3, (1)], we conclude (also see (31.49)) S
PA®B(X
® V) = sVc{x ®y) = spA(x)spB{y),
V (x, y) G H(A) x H(B)
T
and this proves (31.47). For further information on the spectra of elements of a tensor product topological algebra, see [110]. 31.22 Corollary. Let A[TA], B[TB] be unital symmetric Arens-Michael *-algebras and T an inverse limit preserving topology on A® B. (1) // 0 < x\ < X2 in A[TA\ and 0 < y\ < t/2 in B[TB], then one has 0 < x\ (g) yi < X2 ® 2/2) in
A®B. T
(2) I/O < Xi in A[TA\, 0 < yi in B[TB\, i = 1, n and either of A[TA\, B[TB] is commutative, then one has 0 < X^iLi ^i ® 2/i ^n A®B. T
Proof. We first prove (2). Recall that x > 0 in A[TA] means x G -ff(A) with SPA{X) C [0, OO) (cf. Definition 10.10). On the other hand, every symmetric algebra is hermitian (Proposition 21.2(2)). Hence, (cf. (31.47)) Xi ® yi > 0 in A®B, for every i = 1,2,..., n. Now, if either of A[r^], B[TB] T
is commutative, Theorem 34.15 (in Section 34) shows that A®B is also T
symmetric, therefore (see Remark 22.7) J27=i x i ® 2/i ^ 0 in A®B. T
(1) Let 0 < x\ < X2 in A[TA] and 0 < yi < 2/2 in -B[TB]- Then, from the arguments applied at the beginning of the proof of (2) we get that the following inequalities are true in A®B T
Xi®yi>0,
V i = 1,2, xi® (2/2 - 2/1) > 0, (x 2 - z i ) 8> 2/2 > 0,
whence we derive 0 < x\ ® y\ < x\ ® 2/2 < £2 ® 2/2
32. The enveloping locally C*-algebra of tensor products of topological ^-algebras
409
Since every locally C*~algebra is symmetric (cf. (21.6)) we have the following (also see (31.23)). 31.23 Corollary. Let A[TA], B[TB] be unital locally C* -algebras andr one of the C*-convex topologies a, u> (cf. Definition 31.1). (1) / / 0 < x\ < X2 in A\TA\ and 0 < yi < y2 in B[TB\,
one has that
0 < xi ® yi < X2 <8> V2 in A®B. T
(2) 7/0 < Xi in A[TA\, 0 < yi in B[TB\, i — 1,. , n and either of A[TA], n x B[TJS] is commutative, one has that 0 < Y%=i i ® Vi * A®B. T
32
The enveloping locally C*-algebra of tensor products of topological *-algebras
In this Section we study the enveloping locally C*-algebra of a tensor product Arens-Michael *-algebra A®B, with r an inverse limit preserving T
topology on A® B. Our aim is to show that the enveloping locally C*algebra of A®B is expressed (up to a topological *-isomorphism) as the completed tensor product under a and/or u) (see Definition 31.1) of the enveloping locally C*-algebras of A[TA] and B[TB] respectively. For this purpose we need the corresponding result for Banach *-algebras due to K.B. Laursen [245, Theorem 4.8], that we first elaborate. 32.(1) The C*—enveloping algebra of tensor products of Banach *-algebras Suppose that two Banach *-algebras A and B are given together with a *-admissible m-norm n on A ® B (see discussion before Lemma 30.9). Recall that A®B is the Banach *-algebra completion of A®B. Let A and n
n
B possess baVs and consider the enveloping C*-algebras £(A), S(B) of A and B respectively. Endow £(A) ® £{B) with the greatest C*-crossnorm II' Umax an-d let £{A) <S> £{B) be the resulting C*-algebra (see, for instance,
410
Chapter VII.
Tensor products of topological *-algebras
(31.12)). Then, one gets the natural map n
(32.1)
g: A®B —> £(A) ® £{B) with Q{Z) := V ^ ( a * ) ® 0 B ( y i ), n
max
^—' i=l
for every z — YH=ixi ® V%, QA, QB a r e the canonical enveloping maps of A[TA], B[TB] respectively (see (18.11)) and, in fact, g = QA ® QB- Since £ has values in a C*-algebra, it is extendible to all of A®B if and only if n (32.2)
||0(*)||max
V s e A ® B n
(see, for example, [111, p. 9, Proposition]). 32.1 Theorem (Laursen). If A,B are Banach *-algebras with bai's and n a ^-admissible m-norm on A® B that satisfies (32.2), one has that £{A®B) =£{A) n
®£{B), max
with respect to an isometric *-isomorphism. Proof. By (32.2) g is uniquely extended to A®B. Keep the same symbol n
for the extension. Since £(A) <8> £(B) is a C*-algebra, g can be seen as max
a ^-representation of A(8>B. Let a be the *-representation of £(A®B) n
n
corresponding to g. Namely, if Q® : A®B —> £(A®B) is the canonical n
n
enveloping map of A®B, then a o g® = g. Restrict p to g®(A®B). If || || n
n
denotes the C*-norm in £[A®B), we shall show that the restricted a is an isometry on
Q(A®B). n
Let [is be a faithful ^-representation of £iyA®B) and \x the corresponding *-representation of A®B\ that is, jJ-s ° Q® = M- Take the restrictions n
HAiHB of \i to A and B respectively (see Theorem 30.3) and denote by u Ai°~B the corresponding *-representations of £{A) and £(B) such that MA =
v{s) :=
^2aA(xj)aB{yj).
max
32. The enveloping locally C* —algebra of tensor products of topological *-algebras
411
Since a A and o\g commute (see Theorem 30.3), it follows that v is a *representation of £ (A)
\W(x®y)\\<\\aA(x)\\\\oB(y)\\<\\x\\\\y\\,Vx®ye£(A)
®
£(B),
max
therefore by the definition of |[ ||max) one gets that Ms)\\ < ||s|| m a x , V s € £{A) ® £(B);
(32.3)
max
that is v is a continuous ^-representation of £(A) ® £(B), so that it is max
uniquely extended to a *-representation of £ (A) £(B). Keep the same max
symbol for the extension of v. Then, (32.3) is clearly fulfilled for all s € £{A) <S> £{B). It follows easily from the above that fi(z) = (u o ^)(a;), for max
all z G A (g> B ; s o , b y c o n t i n u i t y , [L — VOQ e v e r y w h e r e o n A®B. n
(32.4)
Thus,
n
\\ev{z)\\ = \W(Q®(z))\\ = \\n(z)\\ = MQ(Z))\\
<
\\Q(Z)\UX,
for every z G A®B. To show the inverse inequality, let n® be a faithful n
^-representation of £{A) ® £(B). Then, fi = ii® o Q is a *-representation max
of A®B. Let fi£ be the ^-representation of £lA®B) corresponding to \i. n
n
Then, for every z € A®B, one has EL
(32.5)
||0(*)||max = ||/X®(e(z))|| = ||/Z(Z)|| = | | ^ ( ^ ) ) I I < \\(te(z)\\.
According to (32.4), (32.5), the map a : Q®(A®B) C £(A®B) —> Q(A®B) n
n
n
C £{A) % £{B) max
is clearly an isometry on g(AB'), therefore the assertion follows from the fact that Q
£{A) ® £{B).
n
and Q(A<8>B) is dense in n
m
max
It is easily seen that condition (32.2) is fulfilled by n = || ||7, and in some special cases, it is also satisfied by n = || |[A- Another instance is provided by the next Corollary 32.2.
412
Chapter VII.
Tensor products of topological *-algebras
32.2 Corollary (Laursen). Let A, B be Banach *-algebras with bai's. Suppose that either of them is of type I andn is a ^-admissible m-norm on A®B. Then, the following equality holds up to an isometric * -isomorphism: £(A®B) =
£{A)®£{B). min
n
Proof. Suppose that A is of type /. Then (see Proposition 30.8) the same is true for £(A). Hence, || || max = || ||m;n on £ (A) ®£(B) (Takesaki) (see, for instance, Proposition 31.4). So, according to Theorem 32.1, it suffices to show that n fulfils the condition (32.2). Indeed, suppose that /xi, \ii are faithful ^representations of £(A), £{B) respectively. Then, /ii ^ is a faithful ^representation of £(A) § £(B) [367, p. 208, Theorem 4.9, (iii)]. min
Moreover,
is a ^representation of A®B (cf. Proposition
{HI°QA)®{H2°QB)
n
30.2). So, for z G A®B, one gets n
lleCOIImin = ll(Ml ® 1*2){{QA ® QB)(Z))\\ = ||((Mi ° QA) ® (M2 O QB)){Z)\\ < n(z).
m
32.(2) The enveloping locally C*-algebra of tensor products of m*—convex algebras Take two m*-convex algebras A[TA], B[TB] with baVs and consider their enveloping locally C*-algebras £(A), £(B). Then,
£{A)®£{B) = \im£{Ap) ® £{Bq), (p,q) e TA x TB OJ
*
max
(cf. (31.23) and Theorem 18.11). Now, for every (p,q) G FA x rB, consider the natural (surjective) hornomorphism 6pq
: £{A)®£(B) UJ
—> £(AP) ® £{Bq) = max
{£{A)®£{B))/N^q LJ
and recall the notation of Section 31. For the equality (up to a topological ) in the range of gpq, see (31.18) and Theorems 10.24, 18.11. Then, it is easily seen (by the very definitions of the involved maps) that the diagram
32. The enveloping locally C* —algebra of tensor products of topological *-algebras A®B
e=eA®QB ,
413
£(A)%£{B)
T
U!
Qp0Qq
Qpq
Ap Bq
- £(AP) ® £(Bq)
pq
P
BAv®eBq
max
V
is commutative, where r is an inverse limit preserving topology on A B, with FT — {Tp,q}(p,q)erAxrB
a
defining family of seminorms; gp, gq are the
natural morphisms of A[r^], B[TB] in Ap, Bq respectively and Qi, with i = A,B, Ap, Bq the corresponding canonical enveloping maps. Thus, (32.6)
QPqog=
(gAp ®gBq)o
(gp ® g q ) , V (p, q)e TAx
TB.
On the other hand, it is immediate that if z = X^=i %i® Vi ^ A® B and z
pq = YH=1 xi,P ® Vi,q e AP ® Bq, (p, q) £ FA x TB, the inequality (32.7)
witn
^i.p = xi + ^p> 2/i,? = 2/i + ^9)
wP,,(e(*)) < 7p,,(z), V ( p , ? ) e ^ x r B ,
is equivalent to the inequality (32.8)
\\{gAp (8) toj(^g)||max < H^llp,, V (p,q) e FA x r ^ .
32.3 R e m a r k . It is evident that (32.7), (32.8) ensure continuity of g, respectively gAp ® gBq restricted to A\p\/Np® B[q]/Nq; so that according to pq
(31.17) gAp ® gsq is uniquely extended on Ap®Bq. In conclusion, gAp ® gsq pq
is extendible to all of Ap®Bq if and only if (32.8) holds true. In other words, pq
(32.7) or equivalents (32.8) plays the role of (32.2) (key-tool for the proof of Theorem 32.1) for the Banach *-algebras Av®Bq, (p, q) £ FA x FBpq
32.4 Theorem. If
A[TA\,
B[TB\
are m*-convex algebras with bai's and
T is an inverse limit preserving topology on A 0 B, such that (32.7) (or equivalently (32.8)) is fulfilled, then one has £{A®B)
=£(A)®£(B),
with respect to a topological * -isomorphism.
414
Chapter VII.
Tensor products of topological *—algebras
Proof. Since each one of Afr^], B[TB] possesses a bai, the same also holds for each of the Banach *-algebras Ap, Bq, for every (p,q) G FA x FB. So, according to Remark 32.3, we can apply Theorem 32.1 for the Banach *-algebras Ap®Bq in order to get, up to a topological *-isomorphism, that pq
£{Ap®Bq)
= £{Ap) ® £(Bq),
pq
V (p,q) G FA x FB,
max
Using moreover Theorems 31.9, 18.11, as well as the relations (18.14) and (31.23), we derive £{A®B) = £{\\mAp®Bq) = \im £{Ap®Bq) <
T
pq
<
pq
= ]im£(Ap) ® £{Bq) <
max
= limf(A)p ® £{B)q *
max
= £{A)®£(B), {p,q) GFAx
FB,
up to topological *-isomorphisms. 32.5 Corollary. If A\TA\,
B[TB\
are m*-convex algebras with bai's, r is an
inverse limit preserving topology on A(&B and either of A[TA],
B[TB]
is of
type I, then the following equality holds up to a topological *-isomorphism: £(A®B)
=£{A)®£{B). a
T
Proof. Let A[TA] be of type / . Then, each Banach *-algebra Ap, p G FA, is of type / too (cf. Proposition 30.8). So, applying Corollary 32.2, we take £(Ap®Bq) pq
= £(Ap) § £(Bq), V (p,q) G FA x FB, min
up to a topological *-isomorphism. The assertion now follows by using the same arguments as in Theorem 32.4. 32.6 Corollary. Let A[TA] be an Arens-Michael *-algebra with a bai and Ll(G) the group algebra of a locally compact group G. Let L1(G,A)
be the
generalized group algebra of G. Then, if C*(G) is the group C* -algebra of G, one has the next equality, up to a topological ^-isomorphism: £{Ll(G1A))
= C*{G)®£{A).
32. The enveloping locally C* -algebra of tensor products of topological *-algebras
415
Proof. L1(G) is a *-algebra with a bai [126, (13.4) Theorem]. Moreover, from [262, p. 406] and (31.22) we have Ll(G,A)
= L1(G)®J4 = limZ, 1 (G)§A p , p € FA, 7T
^
7
up to topological *-isomorphisms. According the Remark 32.3 and the discussion after Theorem 32.1, n fulfils the condition (32.7). So the assertion follows from Theorem 32.4 and the fact that £{Ll{G)) = C*(G) (cf., for example, [111, p. 303]). 32.7 Corollary. LetC°°(X) be the Frechet *-algebra of all C°°-functions on a 2nd countable compact n-dimensional C°°-manifold X. Let G be a (non-abelian) locally compact group G and L1(C°°(X),G) the generalized group algebra of G. Then, the equality £(L1(G,COO(X)))
= C(X,C*(G)),
holds up to a topological *-isomorphism. In other words, the non-Banach Frechet *-algebra L1(G,C°°(X)) has a C*-enveloping algebra. Proof. €(C°°(X)) = C{X), up to a topological *-isomorphism (cf. (18.13) and the remarks after it). C(X) is a commutative C*-algebra and since £(Ll{G)) = C*(G) is also a C*-algebra, we conclude (see [367, pp. 211, 215, Theorems 4.14, 4.18] and/or Corollary 31.16) that [| IU = || Umax = || llmin On C(X) ® C*(G) = C*(G) ® C(X). Apply now Corollary 32.6 and [262, p. 391, Theorem 1.1] to get 5(L 1 (G,C OO (X))) = C*(G)®C(X)=C(X,C*(G)). A
32.8 Corollary. Let X be a 2nd countable n-dimensional C°°-manifold and A[rr] a (non-commutative) Frechet *-algebra with a bai and F = {Pn}neN Let C°°(X, A) be the Frechet *-algebra of all A-valued C°°functions on X. IfCc(X,£(A)) is the locally C*-algebra of all £(A)-valued continuous functions on X, under the topology "c" of compact convergence, one has that the next equality holds up to a topological * -isomorphism: £(C'X(X,A))
= CC(X,S(A)).
416
Chapter VII.
Tensor products of topological *-algebras
Proof. From [262, p. 394, (2.8)] C°°{X,A) = C°°(X) 0 A with respect to 7T=£
a topological *-isomorphism. So using the same arguments as in the proofs of Corollaries 32.6, 32.7, we apply Theorem 32.4, (31.23) and (3.23) to get £(C°°(X, A)) = £(Coa(X))®£{A)
= CC{X) § £{A)
u>
e=u>
= limC(K) ® £{An) = i \imC{K)®£{An) < max
A
= CC{X)®£(A) = CC{X,£{A)), n e N. e
32.9 Corollary. Let X be a locally compact space and A[rp] a (noncommutative) Arens-Michael *-algebra with a bai. Let CC(X, A) be the Arens-Michael *~algebra of all A-valued continuous functions on X, with respect to the topology "c" of compact convergence. Then, the next equality holds up to a topological * -isomorphism: £(CC(X,A))=CC(X,£(A)). Proof. Using [262, p. 391, Theorem 1.1] (3.23) and (31.22) we have CC{X, A) = CC{X)®A = limC{K)®Ap,
per,
A
£
where £(C(K)) = C(K), for every compact subset K of X. Denote by id the identity map of C(K) and take the corresponding to (32.8) map id QA : C(K) ®AP
—
C(K) §> £(AP),
A
V p e T,
max
where || || max = || || min — \\ \\\ on C(K) ® Ap (see proof of Corollary 32.6), for any K C X compact and p £ T. Then, id® QAP is continuous (cf. [371, Proposition 43.6]), so its unique extension to C{K)®AV is, in effect, A
a ^-representation, therefore id® QAP fulfils the condition (32.8), hence the condition (32.7) too. Thus, applying Theorem 32.4 and using the same arguments as in the preceding proofs we have £{CC(X, A)) = CC{X)®£{A) = limC{K)®£{Ap) u
<
A
= CC(X)®£(A) = CC(X,S(A)), per.
u
32. The enveloping locally C* —algebra of tensor products of topological *-algebras
417
32.10 Corollary. Let A[rp] be a (non-commutative) Frechet *-algebra with a bai. Let F = {pn}neN and O(C,A) the Frechet *-algebra of all Avalued holomorphic functions on the complex plane C. Then, the equality £(O(C,A))=CC(R,£(A)), holds up to a topological *-isomorphism. Proof. From (3.20) A[rr] = \imAn, where An is the Banach *-algebra, completion of the normed *-algebra A\pn]/NPn. On the other hand, from [262, p. 402, Lemma 4.1], (2.44) and (31.22), the following equalities are topological *-isomorphisms O{C,A) = O{C)®A = limA{TD>n)®An, n e N , e
*
A
where£(A(B n )) = C[-n,n], n e N, therefore£(O{C)) = CC(R) (cf. (18.12)). So the canonical enveloping map of A(H>n), n G N, is gn : A(Bn)
—> C[-n,n] : / .—> /|[_ n , n ] , n 6 N.
Thus, using the same arguments as in the proof of Corollary 32.8, we conclude that the maps Qn ® QAn A{Bn) ® An —> C[-n,n] A
§
S(An),
n e N,
max=A
fulfil the condition (32.8). So, we can apply Theorem 32.4, to get £(O(C, A)) - £{O(C))®£(A) = CC{R)®£(A) = CC(R, £(A)). a;
m
e
A variant of Corollary 32.5 can be given by replacing the inverse limit preservation of r by the weaker concept of *-admissibility, at the cost however of the restriction to m*-convex algebras having a C*-enveloping algebra. As we have noticed, there is a plethora of (non-normed) topological *-algebras that possess a C*-enveloping algebra (see, for instance, Subsection 18.1, Corollary 32.7 and Corollary 32.8 with X compact and A a (non-commutative) C*-algebra). More precisely, one has the following.
418
Chapter VII.
Tensor products of topological *-algebras
32.11 Theorem. Suppose that A[TA], B[TB] are m*-convex algebras that possess a bai. Let either of A[TA], B[TB] be of type I and both have a C*enveloping algebra. Then, if r is a *-admissible topology on A® B, A®B T
also has a C* -enveloping algebra, such that the equality £(A®B)
=£(A)®£(B),
T
a
holds up to a topological *-isomorphism. The proof of Theorem 32.11 is a suitable modification of the proof of Theorem 32.1 (see [155, Theorem 5.13]). Note that according to Proposition 30.8, every commutative locally C*-algebra is of type I. The same is true for the Frechet *-algebra C°°(X), X as in Corollary 32.8, since €(C°°(X)) = CC(X) (cf. (18.13)).
33
The structure space of a tensor product topological *-algebra
In this Section we study the relationship of the structure space of a tensor product Arens-Michael *-algebra to the topological product of the corresponding structure spaces of the participants in the initial tensor product algebra (cf. Theorem 33.15). For this purpose, the decomposition of the enveloping locally C*-algebra of a tensor product Arens-Michael *-algebra given by Theorems 32.1 and 32.4, as well as the relation of the indecomposable continuous positive linear forms of such a tensor product algebra to the corresponding indecomposable continuous positive linear forms associated with the participants in the tensor product (cf. Theorems 33.2, 33.3 and 33.6) play an important role. In the normed case, it was A. Wulfsohn [396] who first, in 1963, studied the decomposition of the structure space of a tensor product C* -algebra. In 1967, K.B. Laursen [245, Theorem 5.5] extended Wulfsohn's result to the case of a tensor product Banach *-algebra. The preceding results are directly related to the corresponding ones of the (non-involutive) commutative case, where the structure space therein is
33.
The structure space of a tensor product topological ^-algebra 419
represented (of course) by the spectrum (Gel'fand space) of the topological algebra involved. Results of this kind have been given by B.R. Gelbaum [165] and J. Tomiyama [370], in the context of commutative Banach algebras, during the period 1959-1960. Later on, in 1964, analogous results to those of B.R. Gelbaum, J. Tomiyama were shown by A. Mallios [255] for non-normed commutative m-convex algebras. Two years later, in 1966, H.A. Smith (cf. [357, Corollary 2]) presented an extension of the results of B.R. Gelbaum, J. Tomiyama for commutative Arens-Michael Q(namely, complete m-convex Q-)algebras, a result contained in those of A. Mallios. In 1968, H.A. Smith considers the analogue of the result of B.R. Gelbaum and J. Tomiyama in the non-commutative case of unital Banach *-algebras, having replaced the maximal ideals with equivalent classes of finite dimensional topological irreducible *-representations (cf. [358, p. 1199, Theorem]). Connections of the above results also exist with relevant results obtained by T. Okayasu [288], T. Okayasu and M. Takesaki, [289], as well as with J. Gil de Lamadrid [170]. 33.(1) Indecomposable positive linear forms Throughout this Subsection A[TA], B[TB) are m*-convex algebras with a bai and r a *-admissible topology on A ® B. Fr = {rK} will denote a denning family of *-seminorms for r and {a\}\&^, {b\}\^A the bai's of A[T^], -B[TB] respectively (see Remark 29.9). Recall that B(A) stands for the nonzero extreme points oiV(A) (see Definition 12.18). We want to see in which way the spaces B(A) x B(B), B(A§>B) are related. In general T
B(A) x B{B) sits inside of B(A<S>B) and under the local equicontinuity of r
both B(A) and B(B), the space B(A) x B(B) is topologically embedded in B(A®B) (see Theorem 33.3). Nevertheless, for "good enough" algebras like locally C*-algebras and the injective tensorial C*-convex topology a in place of r, the spaces B(A) xB(B), B(A®B) become homeomorphic (Theorem 33.5). Results of this kind for Banach *-algebras and C*-algebras can be found in [184, p. 24, Section 5], [279, Section 6.4], [356, p. 23, Chapter 3], [358] and [367, Chapter IV, Section 4].
420
Chapter VII.
Tensor products of topological *-algebras
33.1 Lemma. (1) Each continuous positive linear form h onA®B,
defines
T
a pair (h,A, hg) of continuous positive linear forms on A[TA] and B[TB] respectively, in such a way that tiA{x) — \imh(x ® b\), \/ x & A and hs{y) = limh(a\ ®y), V y G B. A
A
(2) For any continuous positive linear forms f on A[TA] and g on B[TB\ with H/pll — 1 — \\gq\\ (fp, gq are the associated to / , g positive linear forms on Ap, Bq respectively, for some (p, q) e FA X FB), one has that f = {f ® 9)A and g = (f ®g)B(3) Every (/, g) in B(A) x B(B) implies f ® g in B{A®B) and every (f,g) inV{A)xV{B)
with f®g inB{A®B) implies (f,g)
inB(A)xB(B).
Proof. (1) From Theorems 14.2 and 30.3 we deduce that
\imh(x®b\)
= lim < fih(x b\)(£h),£h > = < ( / i / , ) ^ ! ) ^ ) , ^ >, x <E A,
A
A
so lim/i(x ® 6A) exists and does not depend upon the choice of the bai of A
A[r^]. It is now clear that the function tiA
A[TA]
—> C : x i—> HA{X) := lim h(x ® b\), A
defines a continuous positive linear form on A[TA]. In a similar way, one also defines KB(2) Recall that ||/ p || = lim/(a A ) [111, Proposition 2.1.5 (v)], hence U ®9)B{.V)
= iim(/®3)(aA®y) = lim/(oAMy) = \\fP\\g{y) = g(y), A
A
for all y G B. Therefore, (/ g)s = g and similarly (/ (g) g)A = / (3) Suppose now that (f,g) G B(A) x B(B). Then, the corresponding C/V,S-representations /J,J, fig are topologically irreducible (see Theorem 14.6), therefore (cf. Lemma 30.6 and [112, p. 27, Proposition 6]) W* {{pLf ® na){A®B)] = W*[C(HN) ® C(H^g)} =
CiH^H^),
33.
The structure space of a tensor product topological *-algebra 421
that yields /j,f ® \ig £ R'(A®B).
If / (g> # is the continuous positive linear
form deduced by Proposition 30.1, a computation similar to that in the proof of Proposition 30.2 on the equivalence of fi and fii ^2, shows that Hf ® ng ~ Hf®g- Thus (cf. Remark 14.4(2) and Theorem 14.6) / ® g is pure. Then, for some rK G /V (also see Definition 29.5 and Lemma 29.8), I I ( / ® S ) T J | = lim(/®5)(aA®6A)
(33.1) = lim/(a A )limg(6 A ) = ||/p||||ffg|| = 1, A
A
thus we finally conclude that / ® g 6 B(A<S>B). Conversely: Suppose (f,g) G P(A) x P(i?) with / ®g e
B{A®B).
Clearly, / 7^ 0 and j ^ 0. So it suffices to show that / , g are extreme points. Suppose that / 1 , J2 belong to P'(A) in such a way that / = ah + (1 - a)f2, a s (0,1). Tensoring with g and taking into account that / ® g is an extreme point, we clearly derive that (33.2)
h®9 = f®9 = f2®9-
Since ||/ p || < 1 and ||<7g|| < 1 for some (p,q) £ FA X -TB, the equality 1 = ll(/®ff)rj| = ||/P||||5gll ( s e e (33.1)), for some rK G FT, clearly implies H/pll > 1 and || 5 ,|| > 1, so that ||/ p || = 1 = || s ,||. Thus applying (33.2) to the elements x ® b\, A G A, and taking limits, we derive /1 = / = /2- So, / G #(^4) and analogously we show that g G B(B). 33.2 Theorem. (1) The map H : B{A)xB{B)
—> # ( , 4 § £ ) withH(f,g) T
f ® g, for every {f,g) € B(A) x B(B), is a well defined injection. (2) If both A[TA] and B[TB] are unital, the inverse H~l of H on ImH C B(A®B^) is a continuous map. T
(3) If B(A(E)B^ is locally equicontinuous, the following statements hold: (i) H is continuous. (ii) The sets B(A), B(B) are locally equicontinuous.
422
Chapter VII.
Tensor products of topological *-algebras
Proof. (1) From Lemma 33.1(3), H is well defined. To show injectivity of H work exactly as at the very end of the proof of Lemma 33.1(3). (2) Suppose that A[r^], -B[TB] are unital. To prove continuity of H~l on ImH (also see Lemma 33.1(3)), it suffices to prove continuity of ImH —> B(A) (resp. B(B)) : / ® g ,—> / (resp. g). Let ((/ ® g)s) = (fs (g) gs)seA, be a net in Imi7 converging to / ® g G Imi? with respect to the weak*-topology on Z3(.A.B). This means, \im{f5 ® gs)(z) = {f®g){z), d
V z £ A®B; T
so, for z = x®eB, x (z A, we conclude that / is the weak*-limit of (fs)seASimilarly for g. (3) (i) Suppose that B(A®B^ is locally equicontinuous. Let (fs,gs)s^A be a net in B(A) x B(B) converging to (/, g) G B{A) x B(B) with respect to the product of the corresponding weak*-topologies a(A',A), a(B',B) on B(A), B(B) respectively. Then, (fs ® gs)(z) -> (/ g)(z), for all 2; G A 0 B. But continuity is a local phenomenon and B^At&B) is locally T
T
equicontinuous, so using the same arguments as in the proof of Theorem 18.8, we conclude the assertion. (3) (ii) Note that ImH is locally equicontinuous too, so that each f®g in Imf/ (with / G B(A) and g G B{B)) has an equicontinuous neighborhood, say W. Since H is continuous, there are neighborhoods U, V of / and g respectively such that (33.3)
U V C W.
We show that U is equicontinuous. Equicontinuity of W gives that V e > 0 3 a O-neighborhood M in A®U (33.4) with \h'{z)\<£,
V / I ' G V F and z G M.
On the other hand, for g' G V there is y G B with g'(y) 7^ 0, so that by continuity of the map A[TA] —> A®B : x 1—> x ® y, there is a 0neighborhood TV in A[TA] such that x®y G M, for every x e N. Hence (cf.
33.
The structure space of a tensor product topological *-algebra 423
(33.3), (33.4)) \f'(x)\ < k := e/\g'(y)\, for all / ' G U and x € JV. Namely, for every e > 0 there is a O-neighborhood iVo = (e/k)N in J4[TA] such that ]/'(^o)| < £) for all / ' € U and xo G iVo. Thus, U is equicontinuous and in the same way one proves that V also has this property. By the preceding Theorem 33.2, the map H is always denned and it is injective with continuous inverse. Under the assumption of the local equicontinuity of B(A®B), H also becomes continuous (hence, a homeoT
morphism on the image of H in B(A®B)). The next Theorem 33.3 shows that continuity of H can also be obtained under the weaker assumption of the local equicontinuity of the sets B(A) and B(B). 33.3 Theorem. Suppose that A[TA], B[TB] are unital and B(A), locally equicontinuous. Then, the following embedding B(A) x B(B) --> B{A®B)
B(B)
:(f,g)~f®g
is a topological injection. Proof. According to Theorem 33.2, it suffices to show continuity of H. Let (fs,gs)s£A be a net in B(A) x B(B) that converges to (f,g) in B(A) x B(B). Then, we clearly have that (33.5)
{fs gs){z) —» ( / ® g)(z), V
ZGA®B. T
From the local equicontinuity of B(A), B(B) there are equicontinuous neighborhoods U of / in B(A), respectively V of g in B(B) and the set U x V is, of course, a neighborhood of (/,
Proposition 4]. Thus, if W is a neighborhood of f®g in Im(if) c—> S(Aig)B), T
(f7 <8 V) l~l W is an equicontinuous neighborhood of / g, such that cr((U ®V)C\W,A®B)
= a((U ®V)
[78, p. 23, Proposition]. The assertion now follows from (33.5).
424
Chapter VII.
Tensor products of topological *-algebras
In the rest of this Subsection we consider a case, where the map H is also bijective (see Theorem 33.5). For this purpose we first need the following. 33.4 Lemma. (1) An element h in B(A§)B) is of the form hA® hB, if T
either hA G B(A) or hB G B(B) (see Lemma 33.1). (2) Suppose that A[TA], B[TB] are unital C*-convex algebras and A®B a
their injective tensor product (see beginning of Section 31). Then, if either of A[TA], B{TB\
is commutative, one has that for every h in B(A®B) there a
exists (h\, h,2) in B(A) x B(B) with h — h\®h2Proof. (1) Let hA G B(A). Then (see Theorems 14.2, 14.6 and proof of Lemma 33.1), we have that (33.6)
< fJ.hA(x)(thA),thA
>= hA{x) =< {nh)A{x)(Zh),th >,
xeA,
with H&JII = 1 = ||0iJ|. Take the closure (33.7)
K = {{nh)A{x){(,h)
:xeA}
C H^
w i t h fiK(x)
:=
(fih)A(x)\K-
Then, fix is a continuous *-representation of A[r^] in JC(K) (subrepresentation of (HH)A [HI; P- 33]). We show that ^ G K. The *~representation (i^ is non-degenerate as an element of R'(A®B), therefore {/J-h)A has the same property (see, for instance, [184, p. 7, Proposition 1]). Hence (see Lemma 13.4), Oi = limx{iJ,h)A{ax){£h); namely, & G K, therefore (cf. (33.7)) £h is a cyclic vector in K of norm 1. On the other hand, from (33.6) and Lemma 14.5, HhA ~ /j,x and since \x\lA G R'(A) (Theorem 14.6) the same is true for nK (cf. (14.12)). Thus, (HK(A))C
= CidK (cf. Remark 13.3(ii)), while
for any x G A, y G B, the operators (fih)A(X), {fJ>h)B(y) commute (Theorem 30.3(3)). Consequently, (/U/I)B(2/) = \yidx,
y G B, Xy G C, whence
hB(y) = Xy, y e B, and h(x ® y) = < nh(x ® y)(£h), (h >=< (M/I)A(^)(M/I)B(2/)(^)> Zh > = Ay < (fih)A{x)(Sh),Zh > = hA{x)hB(y),
V i G i , y e B.
(2) Suppose that B[TB] is commutative and let Z = (^1®B)C n (A§>i?) be the center of A®B. a
Let /i G B[A§>B) and /i pg the associated to h a
33.
The structure space of a tensor product topological *-algebra 425
element of B(Ap eg Bq) (cf. (12.14), (12.15) and (31.23)). Then, applying min
[367, p. 210, Lemma 4.11] for hpq, we conclude that (33.8)
h{z'z) = h{z')h{z), V z' G A®B and
zeZ.
a
Note that the map B[TB] —> A®B : y — i » e,4 ® y is a topological injection, so commutativity of B[TB] implies B[TB] C Z. Thus, (33.8) yields h(x <8>y) = h((x ® efi)(e/i ® y)) = /i(a; &B)h(eA <E> y), for any x £ A, y £ B. Now, if fti(x) :=ft.(x ® e s ) , V i 6 i and /i2(y) := HeA ®y), V t / e 5 , we conclude that (33.9)
h(x ® j/) = (/ii O /i2)(a; y), V a; 6 A, y e B.
The function hi is by definition a nonzero continuous multiplicative positive linear form, therefore an element of B(B) according to Theorem 14.10. On the other hand, hi G V(A) with hi(eA) = 1 since h G B{A%B).
We show
that hi is also pure. Let / G P(A) with / < hx. Then, / ® h2 < h, where / ® /12 G ^ ( A ® ^ ) . Since /i is pure, there is A G [0,1] such that / ®h2 = Xh (Definition 12.14). This implies / = Xhi, so that hi G B(A) from Proposition 12.16 and (12.10). In view of (33.9), this completes the proof of (2). The following Theorem 33.5, is a direct consequence of Theorem 33.3 and Lemma 33.4(2). 33.5 Theorem. Let
A[TA], B[TB]
be unital C* -convex algebras. Let either
of them be commutative and both sets B(A) and B{B) be locally equicontinuous. Then, for the injective tensor product A®B of A[TA\ and B[TB] we a
have the following equality up to a homeomorphism: B{A®B) = B{A) x B{B). a
33.6 Remarks. (1) For every C*-algebra A, the set B(A) is equicontinuous hence locally equicontinuous. Indeed, the set V(A) is weakly*-compact [111, Proposition 2.5.5, (i)], hence equicontinuous [198, p. 212, Corollary].
426
Chapter VII.
Tensor products of topological *-algebras
(2) There is no (non-normed) locally C*-algebra A[TA] with B(A) equicontinuous, in the sense that a locally C*-algebra A[r^] with B(A) equicontinuous, reduces to a C*-algebra (cf. Corollary 18.24(1)).
The local
equicontinuity of B(A) for a locally C*-algebra A[TA] has no such an effect. Take, for instance, A = CC(R). Then A is a commutative locally C*-algebra, which cannot be a C*-algebra (cf. Theorem 4.26) and its corresponding set B(A) is locally equicontinuous. Indeed, the following equalities (33.10)
B{A) = m(A) = R
hold up to homeomorphisms (cf. Corollary 14.11 and (4.46)). Since A is a Frechet algebra and 9Jt(A) being homeomorphic to R is locally compact, it is also (equivalently) locally equicontinuous (cf. [262, p. 143, Proposition 1.1 and Theorem 1.1]). Thus, from (33.10), B(A) becomes also locally equicontinuous. (3) The next Corollary provides us with a non-commutative locally C*-algebra, which is not a C*-algebra and its corresponding set of nonzero extreme continuous positive linear forms is locally equicontinuous. 33.7 Corollary. Let A be a non-commutative unital C* -algebra. Consider the algebra CC(R, A) of all A-valued continuous functions on R, under the topology "c" of compact convergence. Then, B(CC(R, A)) is locally equicontinuous and the next equality holds up to a homeomorphism: (33.11)
B(CC(R, A)) = R x B(A),
Proof. From [262, p. 391, Theorem 1.1] and Corollary 31.16 we have that CC(R, A) — CC(R) <§> A, up to a topological ^-isomorphism. From Remark 33.6 (1) and (2) £ the sets B(CC(R)) = R (cf.
(33.10)), B(A) are locally
equicontinuous. Hence, the topological identification (33.11) is a consequence of Theorem 33.5. It remains to prove that B(CC(R,A)) is locally equicontinuous. h € B(CC(R,A)).
Then, h = hx®h2
with (huh2)
Let
G B(CC(R)) x B(A) (see
Lemma 33.4(2)), where each one of the /ij's, i — 1,2, attains an equicontinuous neighborhood, say {/,, i = 1,2. Clearly, Ui®U-2, is a neighborhood of h,
33.
The structure space of a tensor product topological *-algebra 427
which is an equicontinuous subset in B(CC(M.) ® ^4) from the *-admissibility of e = a (cf. Definition 29.5). That U\ <8> U A) follows from [76, p. 27, Proposition 4]. This argument also was used in the proof of Theorem 33.3 and originates from various results on the spectrum of a (commutative) topological algebra due to A. Mallios (see, for instance, [262, pp. 146, 410, 411]).
33.(2) The structure space of a tensor product Banach *-algebra If A, B are Banach *-algebras that possess a bai and n is a ^-admissible m-norm o n i ® B , Proposition 30.2 suggests that 1Z(A) x 1Z(B) sits inside of 7?.(A(g)i?), in a natural way. In this Subsection it is proved that the suggested by Proposition 30.2 natural map
(33.12)
G : TZ(A) x 1l(B) —* U{A®B) : ([ Ml ], fa]) >—* fa ® /*2]
is a well defined continuous injection, that under mild conditions becomes a homeomorphism. 33.8 Proposition (Wulfsohn, Laursen). Let A, B, n and G be as before. Then, the following statements hold: (1) G is well defined. (2) G is
1-1.
(3) G is continuous. Proof. (1) Let \J!X e fa] and JJL'2 G fa]- We show that fj,[ //2 e fa <8> fa]Prom the equivalence of /^, Hi, i — 1,2, there are isometric isomorphisms Ui of H^i on H^, i — 1,2, such that Hi(x) oUi=U1o
fj/^x), V x e A and ^{y) oU2 = U2o /i'2(y), V y G B.
Then, U\ U2 is an isometric surjective isomorphism between the Hilbert s p a c e s H^(g>H^2,
H^H^
and
(Mi ® 1^2) (z) ° {U\ ® U2) = (Ui U2) o ( / i i (8) /J,'2)(z),
V z £ A§B;
428
Chapter VII.
Tensor products of topological *-algebras
that is \i\ ® 112 ~ fi[ <S> /x2. Now using the same arguments as in the proof of Lemma 33.1(3) with /J,I, p;2 in place of [if, fig respectively, we conclude that \x\ ® [i2 € R'(A<S>B), consequently (cf. (14.12)) the same is true for n [i[ ® fi'2 a n d [i\ [i'2 € [m [12] G
Tl{A®B). n
(2) Let [fii], [fi'-j] G Tl{A) and [p;2], [/x'2] G 72.(5) such that (33.13)
[/ii (8) /i 2 ] = [/ii ® /i'2] in
Tl{A®B).
Let pi = (/ii (8>/x2)>i, P2 = (M'I ® ^2)B be the restrictions of [i\ ®// 2 , /Uj (8>//2 to
J4[T/I], 5 [ T B ]
respectively. Then, a simple computation based on the
definition of /Oj's, i = 1,2 (Theorem 30.3) and on Lemma 13.4, shows that (33.14)
p\{x) = m(x) ® i^HM2 and Pi(x) = ^ ( x ) ® id// , ,
VxeA
On the other hand, if U is the isometric isomorphism of H^®^ denned by (33.13) and (b\)\€A Pi(x)(U(0)
a bai of
= lim(Aii 8> /i'2)(a: ®
-B[TB],
6A)(C^(0)
on H^i ^^
we conclude that
= U(]im{in
A
® /ia)(x ® 6 A )(0)
A
= C/(pi(x)(C)), V i G A, and £ e
H^^.
This means that pi ~ p'1; which combined with (33.14) yields /xi ~ ^'x according to [111, Propositions 5.3.1, 5.3.3, (ii) and Definition 5.3.2]. In the same way, one gets [12 ~ M21 therefore G is injective. (3) Consider the continuous GiVS-maps (see (14.17)) 5A : B(A) —> ^(,4) : / .—> [/i^ and 5B : 5(B) —^ ^ ( B ) : 5 ^-> [ M |], that in virtue of Theorem 19.7 are also open. Let 5 = 5A X ^B- It is easily seen that 5 is continuous and open too. Consider now the map H : B(A) x B{B) -> S(A®5) : (f,g) >-* f ® g, which is well defined and n continuous according to Theorem 33.2. So the following diagram B{A) x B(B)
^—^
B{A®B) n
ft(yl) x ^(B)
n(A®B)
33.
The structure space of a tensor product topological *-algebra 429
is commutative, where 6® is the corresponding GiV S-map for A®B. Theren
fore 5(g, is continuous and open according to the afore-mentioned. Now, since 8 is continuous and open, one has that G is continuous if and only if G o 8 is continuous (cf. [230, p. 95, Theorems 8, 9]). But, G o 8 is indeed continuous, since G o 8 — 8® o H. 33.9 Proposition (Wulfsohn, Laursen). Suppose that the ^-admissible mnorm n on A® B satisfies condition (32.2). Then, the map G" 1 : ImG —> TZ(A) x 1Z(B) is continuous. Proof. According to Theorems 18.8 and 32.1, continuity of G~l is reduced to that of GJ with G£
K{£(A)) x 1l{£{B)) —> Tl(£(A) §
£(B)).
max
Hence, if pn, pr2 are the projections of 11{£{A)) x TZ{£(B)) on TZ(£(A)) and 1Z{£(B)) respectively, it suffices to show that pr{ o G^1, i = 1,2 are continuous. Let U be open in TZ(£(A)). Then (see Corollary 19.4(2)) there is a closed ideal IQ = 7(e(Cf7)) = n{ker,u : [/z] £ Cf7} in £{A) such that (33.15)
U = {[pi} G TZ(£(A)) : / 0 ^ ker/i, V /x e [A*]},
where Cf7 is the complement of U in 7?.(f (A)). The pre-image of U in 7e(5(A) ® £(B)) is the set V = Geipr^l{U)). Let /i be the closure of max
h ® £{B) in £{A) ® £{B); that is h = Io® £(B)Hm™.
Then, showing
max
(33.16)
V = {[p] G ft(£(A) ® f (B)) : /i % kerp, V p 6 [p]} D ImG f , max
y will be open. If [/t/i®^] € V, we shall have [/ij] G J7, that is (see (33.15)) Ml|/o T^ 0, for every m G [^i]. Hence, m ® ^2\io®s{B) ¥" 0, consequently [/xi ® /X2] belongs to the intersection in the right-hand side of (33.16). Conversely, if [/J^I®^] belongs to the afore-mentioned intersection, we must show that [/ii] G U. Our assumption yields 0 ^ m ® M2|(/0®£(B))[||-||max] = Mil/o ® M2; where /ii|/ 0 <8> ^2 is considered in the sense of Proposition 30.2. Thus, clearly pi|/ 0 7^ 0, which equivalently means [HI] G t/. This completes the proof of (33.16), therefore continuity of pr\ oG^ 1 is shown. In the same way, one proves continuity of pt2 o G^1.
430
Chapter VII.
Tensor products of topological *-algebras
33.10 Theorem (Wulfsohn, Laursen). IfA,B are Banach *-algebras with bai 's and one of them is of type I, then for any ^-admissible m-norm n on A® B, one has the following equality up to a homeomorphism:
n(A®B) = n{A) x n{B). n
Proof. Suppose that A is of type /. Then (see proof of Corollary 32.2) the ^-admissible m-norm n fulfils the condition (32.2). So from Propositions 33.8 and 33.9 the map G (see (33.12)) is a homeomorphism of TZ(A) x U{B) in 72.(A®Z?). The fact that A is of type / also yields surjectivity of G n
according to Theorem 30.11. 33.(3) The structure space of a tensor product Frechet *—algebra As we promised at the beginning of Section 33, we shall study a corresponding decomposition to that of Theorem 33.10 for the structure space of a tensor product Arens-Michael *-algebra. Given an m*-convex algebra A[TT], we have defined a natural topology T$ on the set Tl{A) : = R'(A)/ ~ (consisting of equivalence classes of topologically irreducible *— representations of -A[rf ]) and have called the resulting topological space, structure space of Afrp] (see Definition 14.8 and the discussion before it). Looking at (13.12) and (14.13) we may define the set UP(A) := {\jj] e TZ(A) : [|/x(x)|| < p(x), V x G A}, p e F (33.17)
and verify that: UP(A) = R'p(A)/ ~ = Tl(Ap), V p e T ,
and 1l(A) = {Jnp(A) = \Jn(Ap). In this way, we can prove that one may define on 1Z(A) a second topology, namely the inductive limit topology, that under certain conditions coincides with r<5 (see Theorem 33.14). Based on this fact and using the results of Subsections 33.(1) and 33.(2) we shall show an analogue of Theorem 33.10 for Frechet *-algebras (Theorem 33.15). Results concerning the analysis of TZ(A®B), when A[r^], B[TB] are appropriate m*-convex algebras and r a T
e topology on A ® B, had first been considered by the author in [145], using methods similar to those applied in the Banach *-case by
33.
The structure space of a tensor product topological *-algebra 431
K.B. Laursen (see Subsection 33.(2)). The process we use in this Subsection is different; the corresponding Banach *-algebra results are simply used in a stage, within a natural and particularly flexible approach. Coming back to an m*-convex algebra A[T^] with a bai, consider an element [fip] G 1Z(AP). Then the composition fiq := fip o gpq: p < q in F, defines an element [/j,q] £ lZ(Aq) and the correspondence Rqp : H(Ap) —> n{Aq) : [fip] .—> [fiq}, p < q in F is a well denned continuous map, when TZ(A{) carries the final topology r^, i = p,q, through the respective GiVS-map 5i : B(Ai) — TZ(Ai) : fi H-> [i_ifi], i=p,q. To check continuity of Rqp consider the diagram B(AP)
^
5P
TZ(AP)
^
B(Aq) 5q
—
p
K(Aq)
where for fp e B(AP), (3qp(fp) fq with fq — fp o gpq. The map (3qp, p < q in F, is a well defined continuous injection and the preceding diagram is commutative. So, Rqp o 5P = 5q o (3qp, where 5q o (3qp is continuous. Hence, Rqp is continuous according to the definition of the topology T/>p. Moreover, for p < q < r in F, one has Rrq o Rqp = Rrp. So the family (1Z(AP), Rqp)p
Tl{A) = limft(A p ), p e F.
We shall denote by rn m the inductive limit topology on 7Z(A) (see, for instance, [130, pp. 420-426]).
432
Chapter VII.
Tensor products of topological *-algebras
It is easily seen that (B(Ap), (3qp)p
B{A) = ihnB(Ap); and similarly V{A) = limV{Ap),
p E T.
Denote by TQ the relative topology on B(A) from A's and by r®m the inductive limit topology on B(A) according to (33.19). We shall show that under certain conditions TQ = r^m; for this reason we first need the following. 33.11 Lemma. Let A be a unital Banach *-algebra. Then, the continuous (natural) injection B{A) —> V(A) : f i—> / is closed. Proof. Let V C B{A) be closed. We show that V is also closed in V{A). Let {/[/}, v E A, be a net in V, such that fv —> h, with h E V{A). We prove that h E V. Since, fu E B(A), we get (33.20) h(e) = l i m / ^ e ) = lim ||/,,|| = 1, whence it follows that \\h\\ = 1. V
V
Let now g € V(A) with g < h. Then, for each z e A, we have g(z*z) < h(z*z) — linijy fu{z*z).
Hence, there is an index VQ in A, such that
g{z*z) < fv(z*z*),
V z £ A and v > VQ.
Since each fv is pure, there are numbers au E [0,1] with g = avfu, V v > VQ. Applying in the previous equality a similar argument to that of (33.20), we get that lim,, au — g(e). Thus, g = g(e)h, which together with the last equality of (33.20) implies that h is pure, therefore h E V. Given a topological space X, a family
{5A}AS/1
of compact subspaces of
X is called a k-covering family for X, if for each compact subset K of X there is an index X E A such that K C S\ [262, p. 165, Definition 5.1]. 33.12 Lemma. Let A[rr] be a unital barrelled m*-convex algebra. Then, the family {B(Ap)}per
is a k-covering for B(A).
33.
The structure space of a tensor product topological *-algebra 433
Proof. From (12.16) B{A) = \JBP{A), where BP(A) is homeomorphic to B(AP), for each p G F (see Corollary 12.19). From Lemma 33.11, B(AP) is weakly*-closed in V(Ap) and since V(AP) is weakly*-compact [111, p. 44], B(Ap) is weakly*-compact too. Let now K be an arbitrary weakly*compact subset of B(A). Then, K is an equicontinuous subset of A's [198, p. 212, Corollary], consequently there is a O-neighborhood Up(e) = {x G A : p(x) < e}, 0 < e < 1, in A[T], such that K C C/°(e) = U°V{1), with V = Up(e) and py the gauge function of V. But, p(x) = epv(x), for every x G A, so that Np = Nvv and Ap = APv. Thus, there is p £ F, with X C B(A) n I7p°v(l) = BPV(A) = B(AP) (for the equalities, see (12.15) and Corollary 12.19). This completes the proof. 33.13 Theorem. Let A[rp] be a barrelled m*-convex algebra with a bai. Let B(A) be locally equicontinuous in A'. Then, T$ — r®m on B(A), that is B(A) — lim B(AP), p e F, up to a homeomorphism. Proof. Suppose temporarily that -A[Tf] is unital. Then, from Lemma 3.2, {B(Ap)}p&r is a fc-covering family for B(A). Showing that (B(A),TB) is a A;-space [230, p. 230], we conclude from [262, pp. 166, 167, Lemma 5.2 (B(A),TB) and Corollary 5.1] that TQ = rj?m. It is sufficient to show that is a locally compact space. Indeed, let / G B(A). Since B(A) is locally equicontinuous, there is an equicontinuous neighborhood V of / . By the Alaoglu-Bourbaki theorem, V is relatively weakly*-compact in A's. Now using the same arguments as in the proof of [262, p. 143, Theorem 1.1] we obtain a compact neighborhood of / in B(A). Let us now come to the non-unital case. Passing to the unitization AI[TI] of -A[TT], we have that AI[TI] is a barrelled m*-convex algebra (see Subsection 3.(3) and [198, p. 215, Corollary, (b)]). On the other hand, denning the function fo(x, A) := A for every (x, A) £ Ai, we conclude that /o G B(Ai) and B(A) = B{A\) \ {/o}, up to a homeomorphism that sends an element / G B(A) to an element f\ G B(A\) such that, for any (a;, A) G A\, fi{x,X) := f(x) + A. It is easily seen that B{A{) is locally
434
Chapter VII.
Tensor products of topological *-algebras
equicontinuous, since B(A) has this property. So, as we proved above, B(A\) = limB((Ap)i), up to a homeomorphism. For each p 6 F, the Banach *-algebra (Ap)i, unitization of the Banach *-algebra Ap, coincides (topologically) with the Banach *-algebra (A\)P1 corresponding to the Arens-Michael analysis of Ai[ri] (see proof of Proposition 3.11). Applying now the same arguments as in the proof of [262, p. 169, (6.6)], we are led to the claim of the theorem. 33.14 Theorem. Let A[rr] be a barrelled m*-convex algebra with a bai, such that B(A) is locally equicontinuous. Then, the topologies T$ and ri;m coincide on 7Z(A). Proof. The commutativity of the diagram before (33.18), yields the existence of a unique continuous map lim<5p : \im B(Ap) —> ]im1Z(Ap), p £ F, such that the diagram
is commutative, where ip, j3p are the continuous natural embeddings of n(Ap) in Tl(A)[T[im] and B(AP) in B(A)[T^m] respectively. We show that ip is also continuous when 7Z(A) carries the topology T$. Note that from Theorem 33.13, T^m = rg. On the other hand, if in the preceding diagram we replace ri;m with T$ and lim(5p with 8A (see (14.17)), the diagram remains commutative. Thus, ip : 1Z(AP) —> 71(A)[TS] is continuous if and only if ip o Sp is continuous, which is true since ip o 5P = 5A ° Pp and both 5A and ftp are continuous. Hence, from the definition of Ti;m we conclude that TS -< T\\m. We show the inverse inequality. Let G C 7?.(A)[riim] be open; that is, ipl{G) is open in H(AP), for all p e F. Then 5~1(i~1(G)) is open in B(AP), for all p e F. But, (J-^ip^G)) = /3'1 (S^1 (G)), p e F. Hence, 5A~1(G) nB(Ap) is open in B(AP), for each p G F, which equivalently means
33.
The structure space of a tensor product topological *-algebra
435
that S^1(G) is open in B{A)[r®m = TQ\. Thus, G is r^-open, therefore T
T
lim
6- It follows that r
S
= r
lim
an
d S — lim 8p.
The last equality is due to the fact that lim 5P is the unique continuous map making the preceding diagram commutative. Consider now two m*-convex algebras A[T^], B[T#] and an inverse limit preserving topology T on i ® B. Then, taking into account (31.21) and (33.17) we are led to a corresponding to (33.18) set-theoretical identification for A®B. Namely, T
(33.21)
U{A®B) = lim1l(Ap®Bq),
(p,q) e FA x rB,
pq
T
where the index set F& x Fg takes a preorder by (p, q) < (p', q') if both p < p' and q < q'. So FA x f^ is a directed set and (1Z(AP) x (as well as (7Z(Ap®Bq)),
,
r
x r
1Z(Bq))(p^£pAxrB
)) is an inductive system of topological
spaces, together with the continuous maps (see discussion before (33.18)) Rp'p x Rqlq : 1l(Ap) x TZ{Bq) —> H{Ap>) x 1l{Bq,), with (p,q) < (P',q>) in rA x T B (cf. [130, p. 425, 1.9]). Suppose now that either of the *-algebras -A[TVI], B[TB] is of type / and both have a bai. Let A[TA] be of type / . Then, each Banach *-algebra Ap is of type / (Proposition 30.8). Now, from Theorem 33.10 we have
n(Ap®Bq) = n{Ap) x n{Bq), v {P,q) erAx
rB,
pq
up to a homeomorphism, that we shall denote by Gpq. Note that ([fj,p], [fJ-g]) £ TZ(Ap) x TZ(Bq) goes through G~qx to [fip (g) nq] (where /i p ® /jq also denotes the unique extension of /xp (8) /z9 to A p § i ? 9 ) . Now, from [130, p. 422, 1.5 pq
Theorem], the map (33.22)
G = HmGpq : lim1l(Ap®Bq) ¥
>
pq
—> lim(^(A p ) x H(Bq)) >
436
Chapter VII.
Tensor products of topological *-algebras
is a homeomorphism. Combine now the continuous bijection (see [130, p. 425, 1.9, (3)]) \im(R(Ap) x TZ(Bq)) —> limft(A p ) x \imU{Bq), with (33.22) to get that the map (33.23)
G : limTZ(Ap§>Bq) —> Um1l{Ap) x limH(Bq) y
pq
y
y
is a continuous bijection too. 33.15 Theorem. Let A[TA], B[TB] be Frechet *-algebras, such that each one of them possesses a bai and one of them is of type I. Let r be an inverse limit preserving topology on A®B, such that B(£{A®B)) is locally T
equicontinuous. Then, the following equality holds up to a homeomorphism:
n{A®B) = n{A) x n{B). T
Proof. The local equicontinuity of B(8(A®B)) T
implies that of B(A®B), T
as well as of B(£(A)<E>£(B)), with a the injective tensorial C*-convex topoa
logy (see (31.1), Corollary 32.5 and discussion after Theorem 18.8). In its turn the local equicontinuity of the preceding sets implies local equicontinuity for the sets B(A), B(B) and B{£{A)), B{£{B)) respectively (cf. Theorem 33.2(3)). Thus, from Theorems 33.13 and 33.14, we conclude that the natural topologies of B(A), B(B), B{A®B), Tl(A), Ti(B), Tl{A®B) coincide with their corresponding inductive limit topologies. Therefore, looking at (33.18), (33.19) and (33.23) we get that G : Tl{A®B) —> U{A) x 1l(B) is a continuous bijection. It remains to prove that G" 1 is continuous. For this purpose, consider the following commutative diagram
33.
The structure space of a tensor product topological *-algebra 437
where 5A , 5B, 5® are the corresponding GNS-m&ps, H is denned in Theorem 33.2(1) and G " 1 ^ ] , [//]) = [/*®/i'], for any ([/j], [//]) e 7l(A) x ft(fl). Commutativity of the above diagram gives G~l o (5^ x (5#) = 5®oH, where 8i, i = A,B,®,
are continuous by the definition of the topologies r^, and
H from Theorem 33.2(3). Thus, G~l o (<5^ x <5g) is continuous with 5A X <5# continuous and open (for the last property see Theorem 19.7). Continuity of G" 1 follows now from [230, p. 95, Theorems 8, 9]. There are plenty of non-normed m*-convex algebras, that attain a (normed) C*-enveloping algebra (cf. Subsection 18.(1)). In this respect, we have the following 33.16 Corollary. Let A[TA],
B[TB]
be Frechet *-algebras such that both of
them have a bai and a C* -enveloping algebra and one of them is of type I. Then, for every inverse limit preserving topology r on A ® B, the next equality holds up to a homeomorphism: U{A®B) = 1Z{A) x U{B). Proof. Since £(A), £(B) are G*-algebras, the same is true for £(A(ED-B) (see Corollary 32.5). Thus, B(£(A®B^
is equicontinuous, hence locally
equicontinuous. The assertion now follows from Theorem 33.15. In the sequel we apply the preceding results to some concrete m*-convex algebras. 33.17 Proposition. Let G be a (non-abelian) locally compact group. Let C°°(X) be the Frechet *~algebra of all C°°-functions on a 2nd countable finite dimensional compact C°°-manifold X and L1{G,C0O{X))
the gener-
alized group algebra of G. Then, if C*{G) is the group C*-algebra of G, one has that the following equality holds up to a homeomorphism: (33.24)
n(L\G,
C°°(X))) = ft(G*(G)) x X.
Proof. Note that Ll(G,C°°{X)) = L1 (G)®^*'(X), up to a topological * re
isomorphism, with the projective tensorial topology TT being inverse limit
438
Chapter VII.
Tensor products of topological *-algebras
preserving (see [262, p. 406] and discussion after Definition 31.6). Also note that both of L1(G), C°°(X) have a C*-enveloping algebra (for the second one, see beginning of Subsection 18.(1)). In particular, £(C°°(X)) = C(X), hence £(C°°{X)) as a commutative C*-algebra is of type / [111, 4.2.2, 4.3.1, 5.5.2]. Thus, C°°(X) is of type / , according to Proposition 30.8. Furthermore, since C°°(X) is commutative, its structure space 1Z(C°°(X)) coincides with its topological spectrum (Gel'fand space), which is homeomorphic to X (see (4.43)). The conclusion now follows from Corollary 33.16, provided that H(Ll(G))
= U(C*(G)) up to a homeomorphism (cf. [245, Proposition
2.10] and/or Theorem 18.8). Every m*-convex Q-algebra with a bai, has a C*-enveloping algebra (cf. Corollary 18.16). Thus, in Proposition 33.17, C°°(X) can be replaced, for instance, by any unital Frechet Q-*-algebra A [77-], so that the place of X in (33.24) will be taken by 1Z(A). 33.18 Proposition. Let X be as in Proposition 33.17 and A a (noncommutative) Banach *-algebra with a bai. Let C°°(X, A) be the Frechet *-algebra of all A-valued C°°-functions on X. Then, one has up to a homeomorphism, that n(C°°{X,A))
= X xTZ{A).
Proof. As we have already noticed, from [262, p. 394, (2.8)] we have that C°°(X, A) = C°°(X) <§> A (up to a topological *-isomorphism). According 7T=£
to the proof of Proposition 33.17, C°°(X) is of type / , has a C*-enveloping algebra, TZ(C°°(X)) is homeomorphic to X and TT is an inverse limit preserving tensorial topology. Consequently, the result is again taken by Corollary 33.16. 33.19 Proposition. Let X be a compact space and A[rr\ a unital symmetric Frechet *-algebra, whose the spectral radius is finite on the self-adjoint elements. Let C(X,A) be the unital Frechet *-algebra of all A-valued continuous functions on X. Then, the following equality TZ(C(X,A)) = X xTZ(A) holds up to a homeomorphism.
34- Properties of Permanence
439
Proof. Again we know from [262, p. 391, Theorem 1.1] that C{X,A) = C(X)®A, up to a topological *-isomorphism. Moreover, C(X) is of type / e
as a commutative C*-algebra and A[rr] is a Q-algebra according to Lemma 18.17. The latter fact yields that A[TT] has a C*-enveloping algebra (cf. Corollary 18.16). Clearly, 1Z(C(X)) coincides with the spectrum of C(X), which is homeomorphic to X (see (4.46)). Further, e is an inverse limit preserving topology on C(X) <S> A (see discussion after Definition 31.6), so that the claimed result follows from Corollary 33.16.
34
Properties of Permanence
In this Section we study the heredity of *-semisimplicity and symmetry from a topological tensor product *-algebra to the topological *-algebras participating in the tensor product and vice versa. Questions of this kind, in the context of Banach *-algebras, have been considered by K.B. Laursen in [244, 245]. Earlier R.B. Gelbaum [166] and J. Gil de Lamadrid [170] had faced similar questions concerning semisimplicity in Banach algebras, while A. Mallios [255, 256, 257] and H.A. Smith [358] considered such questions in the more general setting of non-normed topological algebras. It is interesting that in all cases a necessary and sufficient condition for the permanence of (*-) semisimplicity by a tensor product is the faithfulness of the tensorial topology involved. 34.(1) *—Semisimplicity If E[TE], F[TF] are complete locally convex spaces, then Be(Efs, Fj) (see discussion before (29.7)) is a complete locally convex space too (cf. [371, p. 430, Proposition 42.3]). Suppose now that J4.[TA], B[TB] are complete locally convex algebras and r an admissible topology on A ® B such that A®B is a (complete) locally convex algebra (see, for instance, discussion after Definition 29.4). Then the natural continuous linear map (34.1)
iT : A ® B —> Be(A's,B's) : z i—> iT(z) with ir{z){x',y')
:= (x'®y')(z),
V (x',y') e A's x B's
440
Chapter VII.
Tensor products of topological *-algebras
is uniquely extended to a continuous linear map between A®B, Be(A's, B's) T
also denoted by iT. In this respect, we have the following. 34.1 Definition (Mallios). The topology r on A ® B is called faithful, if the continuous linear map iT : A®B —> Be(A's, B's) is injective. T
34.2 Lemma (Laursen). If A, B are C*-algebras, the least tensor C* -norm II ||min on A® B is faithful. Proof. There are Hilbert spaces H\,H2 such that A
> C(H\) and B <—>
£(#2), where <—> means topological injective *-morphism. Thus, A ® i ? M C(Hi®H2)
( f o r HX®H2,
see 29.7). L e t n o w z e 4 ® B min
min w i t h imiri(.z) = 0
(see Definition 34.1). We must show that z = 0. Looking at z as a bounded linear operator on H\<S)H2, it suffices to show that (for <, > see (29.12)) (34.2)
< zfo ® &), 771 <8> % > = 0, V ^ , T?, 6 ffi, i = 1, 2.
Given £,, 77^ G Hi, i = 1,2, define the continuous linear forms /i : £(ffi) —» C : Ti h— ^(Ti) :=< T i t e ) , ^ >, i = 1,2. Then, / = / i | ^ e A' and g = /2IB G S'. Now, it is easily computed that < z(£1®&),Vi®V2
>= (/®ff)(«) = imin(*)(/,fl) = 0 ,
for all ^i,?7i in Hi, i — 1,2; therefore (34.2) is true. 34.3 Remark (Laursen). Lemma 34.2 is not true if || | | m j n is replaced by || ||max- Indeed: Since || | | m j n < || || m a x (cf. Proposition 31.3 and discussion before Definition 31.1), the identity map A ® B —> A <S> B extends uniquely max
min
to the corresponding completions, to a continuous linear map. Denote this extension by i. Then, the following diagram is commutative
34-
Properties of Permanence
441
So if i m a x is injective, the same will be true for i since im-m o i = iiTmax Dealing with C*-algebras, this implies || || m i n = || || max (see, for instance, [111, Proposition 1.8.1]), which is not true, in general (Takesaki [366]). Recall that a topological *-algebra A[T] is called *-semisimple if its *radical R*A is trivial (Definition 18.2). In particular, an m*-convex algebra ] with a bai is *-semisimple if and only if the canonical enveloping map QA A[rr] —> £(A) is injective (Proposition 18.7). The result of the next Theorem 34.4, also due to K.B. Laursen [245, Theorem 6.9], was proved for *-semisimple Banach *-algebras A, B that at least one of the C*-algebras £ (A), £{B) possesses Takesaki's property (T). A C*-algebra A has the property (T) if A B for any min
max
C*-algebra B. M. Takesaki has shown that the class of C*-algebras with property (X1) contains the class of C*-algebras of type / (see [366]). Recall that a Banach *-algebra A with a bai is of type / if and only if its C*enveloping algebra £(A) is of type / (see Proposition 30.8). We prove the afore-mentioned Laursen's result for the less wider class of type / Banach ^-algebras [244, Theorem III1]. 34.4 Theorem (Laursen). Let A, B be Banach *-algebras that possess a bai. Let either of A, B be of type I and n a * -admissible m-norm on A® B. Consider the following statements: (1) A®B is *-semisimple; (2) n is faithful. Then (1) => (2) and under the extra condition that A, B are both *semisimple one also has (2) => (1). Proof. (1) => (2) Take the transpose of the canonical enveloping map g£ : A -> £(A) (see (18.11)). Namely, 'Q^-.^A)'
—> A' :a' ^->
a'ogg.
Let z e A&B with iR(z) = 0. Then (see (34.1)), for x' = a' o g£ e A' and n y' = b' o QB G B' with a' e £{A)' and b' £ £(B)', we have (x' ® y')(z) = 0. If A is of type /, the same is also true for £(A) (Proposition 30.8), so that
442
Chapter VII.
Tensor products of topological *-algebras
II llmin = || Umax on £(A)
Be(£(Ays,£(B)'s)
min
is injective and for every (a',b') G £ (A)' x £(B)', one has (imin o {QJ ® Q^))(z)(a', b') = (a' ® b'){{(£ ® *?#)(*)) = ((a'o^)®(6'oef))(z)=0.
Therefore, (i m i n ° (gg <8 gf))(z) = 0. But, g^ ® gf — IT O g 8 , where 0® is the canonical enveloping map of A®B, hence injective, since A(&B H
Ik
is *-semisimple (see the preceding discussion) and a is the isometric *isomorphism between £(A®B^ and £ (A) ® £{B) given by Theorem H
min=max
32.1. It follows that im-m o (g£ (g> Q^ is injective, consequently 2 = 0. This shows (2). (2) => (1) Suppose that A, B are moreover *-semisimple. To prove (1), according to Proposition 18.7, it suffices to show that the canonical enveloping map g® : A®B —> £ (A&B) is injective. Let 0 ^ z € A®B. n
n
n
From the faithfulness of n, the bilinear form in{z) : A's x B's —> C is nonzero. So there exists {x',y') E A's x B's (cf. (34.1) and (31.38) with Lemma 31.18) such that 0 # in(z)(x\y') = (x' ® y'){z) = y'(^(z)). Hence, 0 ^ tix'{z) & B, for some x' G A's and since B is *-semisimple we derive 0 ^ gf^x^z)) in £(B) (Proposition 18.7). But then, there is b' € £{B)'S with (see (31.39)) (x1 o I9J,/0..B)(.Z) = (b' o Qg o dxi){z) 7^ 0, where x' as before; this in turn yields 0 ^ i)b,0SB(z) in A, with the preceding b'. Since A is *-semisimple, repeating the preceding arguments with A in place of B, we find a1 e £(A)'S such that 0^(a'og£o
dVoQB){z) = ((a' oej) ® (b' o g*))(z) =
{a'®b'){{g$®gB£){z)),
34
Properties of Permanence
443
with (a',b') G £(A)'3 x £(B)'S as above. Consequently (also see proof of the implication (1) => (2)),
and this proves injectivity of g®. 34.5 Remark. Let A[rr] be an m*-convex algebra. Prom the definition of *-semisimplicity (Definition 18.2) and (13.12) one readily verifies that: If each Banach *-algebra Ap of the Arens-Michael analysis of an m*convex algebra A[rr] is *-semisimple, then A[rr] is *-semisimple too. The converse is not, in general, true. So, in several cases, we can derive *semisimplicity of a non-normed m*-convex algebra by just applying Theorem 34.4. On the other hand, in virtue of Theorem 32.11, one can repeat the proof of Theorem 34.4 and get Laursen's result for two (not necessarily complete) m*-convex algebras A[r^], B[TB] that possess a bai, one of them is of type / and both have a C*-enveloping algebra. Recall that there is a plethora of non-normed m*-convex algebras, having a C*-enveloping algebra (cf. Subsection 18.1). More precisely, one has the following. 34.6 Theorem. Let
A[TA], B[TB]
be m*-convex algebras that possess a bai,
one of them be of type I and both have a C* -enveloping algebra. If r is a *-admissible topology on A® B, consider the following statements: (1) A®B is *-semisimple; T
(2) r is faithful. Then (1) => (2) and under the extra condition that both of A[TA] and B[TB] are *-semisimple,
one also has (2) => (1).
We can now see that without any restriction, *-semisimplicity of a topological tensor product *-algebra passes to the topological *-algebras participating in the tensor product (also see [245, Proposition 6.10]); so one has a strong inverse of the situation occurring in Theorems 34.4 and 34.6. 34.7 Theorem. Let
A[TA], B[TB\
be m* -convex algebras that possess a bai.
Let T be a ^-admissible topology on A(&B such that A®B is *-semisimple. T
Then, A[TA] and B[TB] are both *-semisimple.
444
Chapter VII.
Tensor products of topological *-algebras
Proof. From our hypothesis R*.^a = {0}, whence one concludes that for r
a n y (x,y) G A x B w i t h i / O a n d y ^ 0 , x y <£ R * . - n .
Then, we
T
find (see Proposition 18.3) \i e i?(A®_B) such that \i{x ® y) ^ 0. Now, since //(x ® y) — /^A^MsO/) (cf. Theorem 30.3) with /x^ G -R(^) and HB G R(B), one clearly derives /^(x) ^ 0 and fj.s(y) ¥" 0 a n d so according to Proposition 18.3,
J4[T^]
and
-BJTB]
are both *-semisimple.
Combining Theorems 34.6 and 34.7 we get the following. 34.8 Theorem. Let Let one of A[TA],
A[TA], B\TQ\
B[T~B]
be m*-convex algebras that possess a bai.
be of type I and both have a C* -enveloping algebra.
If T is a *~admissible topology on A® B, the following are equivalent: (1) A®B is *-semisimple; (2) r is faithful and
A[TA], B[TB\
are both *-semisimple.
We give now some applications of the above. Before, we need the concept that follows: A locally convex space E[TE] fulfils the approximation property (Grothendieck [180, Chapitre I, p. 165]), if for every precompact subset (cf. E[TE],
[198, p. 145]) K of
-E[T£]
and every 0-neighborhood V in
there is a continuous endomorphism / of
E[TE],
of finite rank, with
f(x) — x € V, for each x € K. 34.9 Corollary. Let X be a locally compact (non-compact) space. Let A be a Banach *-algebra with a bai andCc(X,A)
the (non-normed) Arens-
Michael *-algebra of A-valued continuous functions on X under the topology "c" of compact convergence. Then, CC(X,A) is *-semisimple if and only if A is *-semisimple. Proof. Using the same arguments as in the proof of Corollary 32.9, we derive that when K runs over all compacts in X, then one has (34.3)
CC{X, A) = CC{X)®A = Yim.C{K)®A = limC{K, A).
If CC(X, A) is *-semisimple, then A is *-semisimple from Theorem 34.7. Suppose now that A is *-semisimple. C(K) as a C*-algebra is *-semisimple
34- Properties of Permanence
445
and being commutative is moreover of type / [111, 4.2.2, 4.3.1, 5.5.2]. On the other hand, ||
\\\ is a faithful topology on C(K) (g> A, since C(K)
possesses the approximation property (cf.
[212, p. 412, Theorem 6] and
[180, Chapitre I, p. 168, Corollaire]). Hence (Theorem 34.4), C{K)®A is * A
semisimple, for each K. The assertion now follows from (34.3) and Remark 34.5. The result of Corollary 34.9 remains true if X is a compact space and A[rr] a (non-normed) Arens-Michael *-algebra with a bai and a C*~ enveloping algebra. This can be taken as an application of Theorem 34.8. 34.10 Corollary. Let X be a compact 2nd countable finite dimensional C°° -manifold and A[rr] a (non-normed) Frechet *~algebra with a bai and a C*-enveloping algebra. Then, the Frechet ^-algebra C°°(X, A) of all A-valued C°°'-functions on X is *-semisimple if and only if A\rr\ is * semisimple. Proof. C°°(X, A) = C°°(X)®A up to a topological *-isomorphism [262, TV
p. 394, (2.8)]. If C°°(X, A) is *-semisimple, then A[rr} is *-semisimple from Theorem 34.7. Suppose now that
J4[TT]
is *-semisimple. Since £(COC(X)) =
C(X) with respect to a topological *-isomorphism (cf. (18.13) and comments after it), the canonical enveloping map of C°°(X) is injective, so C°°(X) is *-semisimple (Proposition 18.7). The same identification shows that C°°(X) has a C*-enveloping algebra. In addition, C°°(X) is of type / , since (its enveloping C*-algebra) C(X) has this property (see proof of Corollary 34.9 and Proposition 30.8). Finally, C°°(X) is a nuclear space, so = e on CCO{X)®A and £ is a faithful topology (cf. [180, Chapitre II, p. 54 ff., as well as Chapitre I, p. 169, Lemme 19] and [212, p. 483, Corollary 2]). Apply now Theorem 34.8 to get *-semisimplicity of C°°{X, A). 34.11 Corollary. Let A be a (non-commutative) Banach *-algebra with a bai and the approximation property. Z/eiC°°(R", A) be the Frechet * -algebra of all A-valued C°°-functions on W1. Then, C°°(Rn, A) is ^-semisimple if and only if A is ^-semisimple.
446
Chapter VII.
Tensor products of topological *-algebras
Proof. Using (2.21), (31.20), [371, p. 449, Theorem 44.1] and similar arguments as those in the proof of Corollary 34.10, we derive
C°°(R",A) = C°°(Rn) ® A = \imC{m)(Km)®A = 7T=£
*
A
<
limCim)(Km,A),
where C^ (Km) is the Banach *-algebra of all continuously m-differentiable functions on the interior of the compact subset Km of IRn, m e NUO. Each C^m\Km) is a symmetric algebra (cf. Proposition 21.6), since each of its characters (see (2.22), together with (4.44) and (4.45)) preserves involution. Thus, C(m\Km), m 6 N U 0, being semisimple is *-semisimple (see [327, Theorem (4.7.15)] and/or Proposition 22.21). On the other hand, C w ( i f r a ) , m £ N U 0, is of type /, since C°°(Rn) has this property (cf. proof of Corollary 34.10 and Proposition 30.8). Finally, || \\\ is faithful on Cm(Km) ® A, m & N U 0, because of the approximation property of A (cf. [180, Chapitre I, p. 168, Corollaire]). So if A is *-semisimple, C(m)(-RTm)®A A
is *-semisimple, for all m G N U 0, according to Theorem 34.4. The rest of the proof follows from Remark 34.5 and Theorem 34.7. A locally compact group G is called of type I, if the C*-algebra C*(G) of G (that is the enveloping C*-algebra of the group algebra L}{G)) is of type / [111, p. 303, 13.9.1, 13.9.3]. 34.12 Corollary. Let G be a locally compact group of type I and A[rr] an Arens-Michael *-algebra with a bai and a C* -enveloping algebra. Then, the generalized group algebra L 1 (G, A) of G is *-semisimple if and only if A[rp] is *-semisimple. Proof. According to the proof of Corollary 32.6 we have the topological *-identification Z/1(G, A) — Ll(G)®A, where LX{G) is a Banach *-algebra IT
with a bai. Moreover, since G is of type I, Proposition 30.8 implies that Ll(G) is of type 7, while from [327, p. 323] LX(G) is *-semisimple. On the other hand, Ll(G) satisfies the approximation property, so that TT is faithful on Ll(G) A [180, Chapitre I, p. 168, Corollaire]. The assertion now follows from Theorem 34.8.
34- Properties of Permanence
447
34.13 Corollary. Let G be a locally compact group and A = C°°(X) with X a compact 2nd countable finite dimensional C°° -manifold. Then,
Ll{G,A)
is *-semisimple. Proof. In the proof of Corollary 34.10 we noticed that C°°{X) is of type / . For the rest of the proof, we just argue as in the proofs of Corollaries 34.10 and 34.12, in order to apply Theorem 34.6. Concerning *-semisimplicity of Banach vector-valued group algebras, see [245, p. 483, Section 7]. In the proof of Corollary 34.12 it was mentioned that L1(G), with G a locally compact group, is a *-semisimple Banach *-algebra with a bai and the approximation property, so a direct consequence of Corollary 34.11 is the following. 34.14 Corollary. Let G be a locally compact group and A = L1(G).
Then,
n
the Frechet *~algebra C°°(R , A) is *-semisimple. 34.(2) Hermiticity and symmetry In the commutative case both concepts, hermiticity and symmetry, coincide on every involutive Arens-Michael algebra (Theorem 21.10). We recall that, in the context of non-normed involutive topological algebras, hermiticity coincides with symmetry in the class of spectral involutive Arens-Michael algebras that contains all involutive Arens-Michael Q-algebras (cf. Theorem 22.27 and Corollary 22.28). The next theorem gives conditions under which hermiticity and symmetry of two Arens-Michael *-algebras passes to their tensor product with respect to an inverse limit preserving tensorial topology and vice versa. In the case of Banach *-algebras (where hermiticity coincides always with symmetry according to the the Smrali-Ford Theorem (see Theorem 22.23)) such a result has been proved by K.B. Laursen [244, p. 65, Theorem III.3]. 34.15 Theorem. Let A[TA],
B[TB]
be unital Arens-Michael *-algebras and
T an inverse limit preserving topology on A® B. Consider the following: (1) A®B is hermitian (respectively symmetric).
448
Chapter VII.
Tensor products of topological *-algebras
(2) A[TA], B[TB) are hermitian (respectively symmetric). Then (1) =*> (2), while (2) => (1) when moreover either of A[TA], B[TB] is commutative. Proof. (1) => (2) Without loss of generality we may suppose that P{CA) = 1 = q{es), for any (p, q) e FA X FB (see Theorem 2.3). Let {r Pig } be a defining family of *-seminorms for r. Then (see discussion before (31.11)), r is a Hausdorff topology and Tp,q{x®eB) = p(x)q(eB) = p(x), Vx G A, resp. TPA(eA®y) — q(y), V y e B , for all (p,q) £ FA X FB- Hence, the maps A[TA]
(resp. B[TB\)
—> A®B : x (resp. y) i—> x®es
(resp. e^ ® y)
T
are clearly topological injective *-morphisms, by means of which A[TA\ and B[TB] become closed *-subalgebras of the hermitian (respectively symmetric) Arens-Michael *-algebra A®B. According to Proposition 21.3, this T
implies that J4[T,4] and Bfrg] are hermitian (respectively symmetric). (2) => (1) Suppose that A[r^] is also commutative. Then (see Lemma 31.18(2)) fly is a continuous *-morphism on A®B, for all ip in UJl(A); note T
that hermiticity coincides with symmetry in all commutative involutive Arens-Michael algebras (see Theorem 21.10 and/or Proposition 21.6). So hermiticity (respectively symmetry) results easily from Theorem 31.20(2), since B[TB] is hermitian (respectively symmetric). For the symmetry use the characterization given by Proposition 21.2(3). 34.16 Corollary. Let X be a locally compact space and A[rr] a unital Arens-Michael *-algebra. Let CC(X,A) be the algebra of all A-valued continuous functions on X, with the topology "c" of compact convergence. Then, CC(X,A) is hermitian (respectively symmetric) if and only if A[rr] has this property. Proof. CC(X,A) = CC{X)®A, with respect to a topological *-isomorphism e
[262, p. 391, Theorem 1.1], where the biprojective tensorial topology e is an inverse limit preserving topology on CC(X) ® A (see discussion after Definition 31.6). At the same time, CC(X) is a commutative unital locally
34- Properties of Permanence
449
C*-algebra (cf. Example 7.6(3)). As a locally C*-algebra, CC(X) is symmetric (cf. (21.6)), hence hermitian. So the assertion follows from Theorem 34.15. Further applications of Theorem 34.15 are given by the Examples 21.17 (4) and 21.17(6) that refer to vector-valued C°°-functions and the generalized group algebra of a discrete locally compact abelian group G. The symmetry of the generalized group algebra Ll{G, A), when G is a compact group and A an arbitrary Banach *-algebra has been investigated by K.B. Laursen [245]. More precisely, K.B. Laursen has proved in [245] that Ll(G, A) is symmetric if and only if A has this property. D. Birbas has studied the symmetry of the generalized group algebra Ll{G, A), when G is a locally compact group and A[rr] an Arens-Michael *-algebra. Among others he proves that L1 (G, A) is symmetric if and only if A[rr] and all closed *-ideals of L 1 (G, A) are symmetric [63, p. 249, Theorem 19.12]. He also considers the case when G is a compact group and A[rp] an Arens-Michael Q-*-algebra and he shows that Ll{G, A) is symmetric if and only if the so-called S-radical of L 1 (G, A) is closed and A[Tr] is symmetric [63, p. 259, Corollary 20.7]. The extension of the afore-mentioned result of K.B. Laursen by D. Birbas, has been achieved by an essentially different process in which "S'-radical" (introduced by J. Wichmann [390]) plays an important role.
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This Page is Intentionally Left Blank
Index C^[0,1], 19
A[T]/I,
40
fif, 167
A[Tr] = lim A[p]/ATP = lim^ p , 16
( , 4 , / i , ^ ) , 160
A[rr] ^
\hnA\p]/Np
*-s&-algebra, 251
<~> lim^ p ,
16
e m-norm, 382
^4[TC*],
e topology on A B,
A\p], 5
212
368
A\p]/Np, 5
, 33
A§>B, 386
*-ideal, 31
^g^^
*-morphism, 31
^
386 3m
T
*-norm, 29
A®B, 367, 368 ^-preserving m-semmorm, 29 g character, 54 g map, 31 on g semmorm, 29
r
A+, 125 Ailn], 39 Ab
, 210
'
132
A
5 ' B(X), Borel cr-algebra, 320 ' C*(G), group C*-algebra of G, 414 p
, 160 on
, 29 -semisimple algebra, 210
£ F
, 31
f '
l algebra, 86
E
®F> 3 6 7
E®F, 367
y bounded algebra, 251
E', topological dual of E, 9
, 29
E'a = E'\o{E', E)}, 9
0-dimensional space, 77
ES(A), 305
< , >, inner product, 9
E\p], 3, 4
A[T], 7
E\p]/Np, 4 487
488
Index
E®F, 389
R'(A), 162
EIF,
R'(A\p]/Np), 163
389
A
^ £®F, EVF, 369
R\AP), 163 ^(i4)) 163
£"*, algebraic dual of E, 4
R(A) 162
£
i?(A[p]/p), 162 R(A ) 162
P' 4 GqA, quasi-invertible elements of A
' 43 GA, invertible elements of A, 42
R*A, *-radical of A, 210 ^ /^-j ^g2
i/, Hilbert space, 9
Rep{A), 305
H(A),
S'(A) 204
self-adjoint elements of A, 28
H
{U),
^W-
S(A), 145 5(R n ), 23
91 14
5^, 320
Hi®H2, 371
£7(^4); unitary elements of A, 28
H
i*i
160
[/*-seminorm, 280
h
166
J7°(e), 54
R
^ m > 239
C/p(l), 5
^ r ( A ) , 305
[;p(e))
J^l, Jacobson radical of ^4, 60
W*[A], 379
Kp = Tl(A) n I7p°(e), 116
xc, commutant of X, 51
^(G),
X cc , bicommutant of X, 52
30
8
L^G, A), 137,414 L p -norm, 14
X/, 165 ^(21), 379
L w [0,l],14
C[[*]], 90
i / , 165
CN, 5
M(X), 320
CH,
M C (X), 320
D n , n-disc, 25
N(A),
$, tensor map, 366
normal elements of A, 28
Np = ker(p) := p-^O},
4
379
r = {p}, 7
ivfp, 218
r~r',
8
P'(A), 145
^-preserving transformation, 333
P(A), 145
A = { P l }, 39
Index
489
rA = {P},7
267, 325
JV, 365
-M[0,l],9
rT = { r j , 368
O(C), 25
r£, 369
CG, complement of G, 240
^4/7 = {«}, 40
V(A), 148
Tr =
{TK},
419
5 A , 172
N, natural numbers, 5
§
K, real numbers, 3
<# 379
a
2)(Mn), 21
a
> { p,q}(p,q)erAxrB,386
f 212 P>
336
2>,(G), 89 154
^(^)> C(X), 29
SW(A), (topological) spectrum of Aj 52
C°°[0,1], 18 C°°-topology, 18
Tl*(A), hermitian spectrum of A, 54
C (M ), 19
(j^^ complex numbers, 3
{m)
C (Km),20 CC{X)+, 323
M ~^,168 M l w M2) 3 8 0
C
°(X)'29 ^ ( ^ ( A O ) , 119
C
Mi®M2, 372 MA; 3 7 7
29
^^)'
MB,
CcPO 35
377
w, H,a( Pl9 ) e ^xr B ,386
'
Ce(m(A)), 116
5
^ = — , j-partial derivative, 19 5(A), 212
^
f(A) p , 218
"<, 12
^, Gel'fand map, 53
^(^)>
/C, 116
{^p.Jcp^er^xrs- 365
ftpO*),
171
241
£(77), 9
^(^',-B), 9
£JPQ-algebra, 353
supp(/), 22
£^~algebra, 352
n , 39
CS(H), 11
r, 7
Cw(H), 9
r^, 24
A^(A), algebraic spectrum of A,
r^, 22
490
Index
TS, 24
ei = (0,1), identity of Ax, 38
rr~rr/,8
/®5,
374
fc
rr, 7
/( ), fc-derivative of / , 18
ro, 25
/L
156
TS,
172
/ p , 145
TS,
22
/ ^ , 164
Too, 18
f®g, 375
r e , 239
5
r e , 116
/c-covering family, 432
TS, 11
p A , Ptak function, 266, 279
TW, 9
P(7, gauge function, 6
TToo, 20
TA{X),
< / , 151
spectral radius of x in A,
Tlhn, 431
46
rc», 211
r p , 208
Re I, real part of I, 324
sp^(x), spectrum of x in A, 43
£>P, 15
uA, 280
gf, canonical enveloping map, 213
x + A = (x, A), 38
Qf,213
x o y , 42
^Pq, connecting maps, 16
x°, quasi-inverse of x, 43
$ x ', 402
x" 1 , inverse of x, 42
e-semiball, 8
a;1/2, square root of x, 127
£
i {ep,?}(p,?)erj!xrPi 369
xf = x + Lj, 176 X p = x + -/Vp, 4
A, Gel'fand transform algebra, 54
n, 382
T, 8
P(A), 146
A[TP],
8
p p (A), 147 Z\p, 336
Zf, 167 r
{ p,?}( M )6r E xr F , 389
Prim(A), 239
ai, approximate identity, 135
(x, A), 38
bai, bounded approximate identity, 135 c, 35 e and/or eA, identity of A, 4
absolutely homogeneous nonnegative real function, 3 additive transformation, 333
Index admissible tensorial C*-convex topology, 368 admissible tensorial m-convex topology, 367 admissible tensorial m*-convex topology, 368 admissible tensorial locally convex *-topology, 368 admissible tensorial locally convex topology, 367 admissible topology onA®B, 367 advertibly complete algebra, 69 algebra, 4 algebra with involution, 27 algebraic dual, 4 algebraic spectrum, 267, 325 algebraically irreducible *-representation, 161 algebraically irreducible representation, 60 approximate identity, 135 approximation property, 444 Arens-Michael *-algebra, 31 Arens-Michael algebra, 13 Arens-Michael analysis, 16 Arens-Michael decomposition, 16 associated *-representation, 163 associated positive linear form, 146 B-complete algebra, 41 B-complete space, 41 Banach *-algebra, 29 Banach algebra, 5 Banach space, 4
491
barrel, 7 barrelled locally convex algebra, 7 basic 0-neighborhood, 8 bicommutant, 52 biprojective tensorial topology, 369 Borchers algebra, 36 bounded approximate identity, 135 bounded part, 132 C*-algebra, 102 C*-convex algebra, 102 C*-enveloping algebra, 222 C*-norm, 101 C*-property, 101 C*-seminorm, 101 C*-seminormed algebra, 101 C*-spectral algebra, 86 canonical bilinear map, 366 canonical enveloping map, 213 character, 52 circle operation, 42 commutant, 52 commutative subset, 80 compact Stein set, 356 compatible involution, 358 compatible tensorial locally convex topology, 367 compatible topology on E®F, 366 condition (L), 337 connecting maps, 16 continuous ^-representation, 160 cyclic ^-representation, 161 cyclic vector, 161
492
defining family of seminorms, 7 enveloping locally C*-algebra, 212 equivalent ^-representations, 168 essential left Frechet A-module, 187 extendable positive linear form, 156 extended seminorm, 236 extreme point, 152 factor ^-representation, 379 factor von Neumann algebra, 379 faithful *-representation, 194 faithful tensorial topology, 440 field algebra, 36 Frechet *-algebra, 31 Frechet C*-convex algebra, 102 Frechet Q-*-algebra, 69 Frechet Q-algebra, 69 Frechet algebra, 13 Frechet locally C*-algebra, 102 Frechet locally convex *-algebra, 31 Frechet locally convex algebra, 7 Frechet topological algebra, 6 fundamental 0-neighborhood system, 7 gauge function, 6 Gel'fand map, 53 Gel'fand topology, 53 Gel'fand transform, 53 Gel'fand transform algebra, 54 Gel'fand-semisimple algebra, 274 global spectrum, 53
Index GNS-construction, 164 GNS-map, 172 GNS-representation, 168 group of invertible elements, 42 group of quasi-invertible elements, 43 group of type /, 446 Hausdorff completion, 4 Hausdorff locally convex topology, 8 hemicompact space, 36 hermitian algebra, 267 hermitian character, 54 hermitian positive linear form, 144 hermitian spectrum, 54 hermitian topological *-algebra, 267 hermitian topological algebra, 267 Hilbert space tensor product, 371 ideal, 5 idempotent subset, 5 indecomposable positive linear form, 152 injective tensorial C*-convex topology, 386 injective tensorial topology, 369 inverse limit preserving tensorial topology, 390 inverse limit preserving topology on A®B, 391 inverse limit preserving topology on E®F, 391 invertible element, 42
493
Index involution, 26 involutive m-convex algebra, 31 involutive algebra, 27 involutive Arens-Michael algebra, 31 involutive locally convex algebra, 31 involutive topological algebra, 29 Jacobson topology, 239 k-space, 35 left ai, 135 left Frechet A-module, 187 left invertible element, 42 left quasi-invertible element, 42 linear involution, 27 locally C*-algebra, 102 locally convex *-algebra, 30 locally convex algebra, 7 locally equicontinuous subset, 55 locally Hilbert space, 106 lower semicontinuous, 236 Lumer group of transformations, 333 m*-barrel, 33 m*-convex algebra, 30 m*-norm, 29 m*-seminorm, 29 m*-seminormed algebra, 29 m-barrel, 7 m-barrelled algebra, 13 m-convex algebra, 13
m-convex topology, 13 m-norm, 5 m-seminorm, 5 m-seminormed algebra, 5 Michael topology, 116 Minkowski functional, 6 modular ideal, 59 modular left ideal, 59 morphism, 15 multiplicative subset, 5 multiplicative transformation, 333 n-disc, 25 non-degenerate *-representation, 161 nonnegative real function, 3 norm, 4 normal element, 28 normal subset, 49 normed *-algebra, 29 normed algebra, 5 normed space, 4 orthogonal basis, 228 positive element, 125 positive linear form, 144 positive measure, 323 pq-(*-)admissible tensorial topology, 389 pq^compatible tensorial topology, 389 pre-C*-algebra, 102 primitive ideal, 239
494
projective tensor product, 366 projective tensorial C*-convex topology, 386 projective tensorial topology, 365 Ptak algebra, 41 Ptak function, 266, 279 Ptak inequality, 266, 280 Ptak space, 41 Ptak type inequality, 285 pure positive linear form, 152 Q-algebra, 69 quasi-equivalent ^representations, 380 quasi-invertible element, 43 quasi-square root, 64 radical, 60 radical algebra, 60 representable linear form, 175 representation, 59 restriction of [A®B, ii}H^), 378 Riesz representation theorem, 320 right ai, 135 right invertible element, 42 right modular identity, 59 right quasi-invertible element, 42 S-radical, 449 saturated family of seminorms, 7 sb-algebra, 95, 251 self-adjoint element, 28 self-adjoint subalgebra, 27, 29 seminorm, 3
Index seminormed space, 3 semisimple algebra, 60 sequentially complete, 197 spectral C*-seminorm, 86 spectral m*-seminorm, 86 spectral algebra, 86 spectral radius, 46 spectral seminorm, 86 spectrally bounded algebra, 95, 251 spectrum of a topological algebra, 53 spectrum of an element, 43 square root, 64, 127 ssb-algebra, 249, 252 state, 144 Stein manifold, 356 strictly positive element, 125 strong operator topology, 11 strong spectrally bounded algebra, 252 strongly-semisimple, 275 structure space, 172 subadditive nonnegative real function, 3 submultiplicative nonnegative real function, 5 submultiplicative seminorm, 5 subrepresentation, 424 support, 21 support of a measure, 320 symmetric algebra, 224, 267 symmetric involution, 267 symmetric topological *-algebra,
495
Index 224, 267 symmetric topological algebra, 267 To-space, 240 tensor map, 366 topological *-algebra, 29 topological *-isomorphism, 31 topological algebra, 6 topological dual, 9 topological injective *-morphism, 31 topological injective morphism, 15 topological isomorphism, 15 topological spectrum, 53 topologically irreducible ^representation, 161 topology of compact convergence, 35 trivial *-representation, 161 type / *-algebra, 379 type / ^-representation, 379 type / m*-convex algebra, 379 type / von Neumann algebra, 379 uniform m-convex algebra, 94 uniform seminorm, 94 uniformly bounded left ai, 187 unit semiball, 5 unit vector, 184 unital algebra, 4 unitary element, 28 unitization of an algebra, 38 variation of a measure, 320
von Neumann algebra, 379 weak C*-property, 101 weak operator topology, 9 weak topological dual, 9 weak*-topology, 9 weakly positive linear form, 293 zero functional, 55
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