A characterization of a class of locally compact Abelian groups via Korovkin theory

Math. Z. 204, 451M64 (1990)

Mathematische Zeitschrlft @)Springer-Verlag1990

A Characterization of a Class of Locally Compact Abelian Groups via Korovkin Theory Michael P a n n e n b e r g Mathematisches institut der Westf/ilischenWilhelms-Universit/it, Einsteinstrasse 62, D-4400 Miinster, Federal Republic of Germany

1. Introduction

The famous duality theorem of Pontryagin and van Kampen shows that every topological and algebraic property of a locally compact abelian group G corresponds exactly to a dual topological or algebraic property of its character group X = G, since X determines both the algebraic and topological structure of G = R. This correspondence has been intensively studied in abstract harmonic analysis; we refer the reader to [H-R], [Ru], [Po], [We] and [Mor] for a survey as well as for proofs of the facts used in the sequel and only mention one example: Let G be an abelian group with discrete topology and compact character group X = G. Then a theorem of Pontryagin asserts that the (Lebesgue covering) dimension of X is equal to the torsion-free rank r0(G) of G, so that we get in particular the following equivalence: G has finite torsion-free rank ~ X has finite dimension

(1)

Another device in the study of a locally compact abelian group G is the consideration of the associated group algebra L I(G), which also completely reflects the structure of G. Having this in mind, it seems desirable to characterize a property of L 1(G) in terms of properties of G as well as dual properties of X = G. We adopt this point of view to solve the following problem which naturally occurs in the theory of Korovkin approximation in commutative Banach algebras (JAIl, 2, 3], [Pa6]): Characterize those locally compact abelian groups G for which the commutative Banach algebra L 1(G) possesses a finite universal Korovkin system. Here a subset T of a commutative Banach algebra A with a continuous symmetric involution * is said to be a universal Korovkin system, iffthe following analogue of the classical Korovkin theorem ([Ko 1, 2], [Ba]) is true: For every commutative Banach algebra B with continuous symmetric involution, every *-algebra-homomorphism L: A ~ B and every net L~: A---*B

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of positive contractions, the convergence p(L~x--Lx)~O (xET) already implies p(L~y-- Ly) ~ 0 (yeA), compare definition (4.2) of EAI3]. Here p denotes the spectral radius of B. We refer to [A11-4], [-Ba], [-Be], [B-L] and [-Pa 1-6] for a series of equivalent definitions and more background on the classical Korovkin theorem and some of its subsequent generalizations. If G is an arbitrary locally compact abelian group (LCA group for short), the fact that LI(G) possesses a finite universal Korovkin system is equivalent to the following property (P) which will be crucial for the sequel: (P) There exist finitely many functions f l . . . . . f,~L~(G) whose Fourier transforms f l , . . . , f , strongly separate the points of X = G, i.e. have the following two properties, (P 1) f l . . . . ,f~ separate the points of X = G, (P2) For every xEX there exists l < k < - n such thatfk(Z)oe0. One implication is explicitly proved in Corollary 4.6 of [-A13] ; the other implication obviously follows from Theorem 4.1 of [-A13]. The paper is organized as follows: The second section just settles some notations. In the following two sections, we characterize the fact that L~(G) possesses a finite universal Korovkin system by establishing a characterization of those LCA groups G which have property (P): It turns out that these are exactly those LCA groups whose character group is a finite-dimensional separable metric LCA group, or equivalently, exactly those groups for which a certain discrete factor group has finite torsion-free rank. The last section of the paper contains a vector-valued version and some comments and corollaries.

2. Notational Conventions and Topological Preliminaries To avoid ambiguity, we first clarify the terminology we will use. All topological spaces under consideration will assumed to be Hausdorff, and we shall consider only second countable locally compact groups. There are several useful ways to define the dimension of a topological space - we refer the reader to [El for their definitions and relevant properties. Since locally compact groups are second countable iff they are separable metric, it turns out that for this class of locally compact groups all these definitions coincide - nevertheless, we will only use Lebesgue's covering dimension and refer the reader to [C], [El, [H-W], [Ku], [M-Z], [Na], [Ne], [Pas] and [We] for a survey of different notions of dimension and the proof of their equivalence for second countable locally compact groups. Finally, the definition and relevant properties of the torsion-free rank r 0 (G) of an abelian group G are to be found e.g. in [H-R], [Po] and [Fu].

3. The Case of a Discrete Group We begin our investigations with the case of a discrete group - this clearly reveals the particularities of this special case and prepares the ground for the proof of the general case in the subsequent section.

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Proposition 1. Let G be a discrete abelian group with compact character group

X = t~. i) ii) iii)

Then the following assertions are equivalent. L 1(G) possesses a finite universal Korovkin system X is a compact metric space of fin~te dimension G is a countable group of finite torsion-free rank

Proof i)~ii): Let {ft . . . . . f.} be a finite universal Korovkin system for L 1(G). Then the Gelfand transforms of these functions separate the points of the maximal-ideal-space of L~(G) (FA12], FPa2]). Identifying the latter space with the dual group X and the Gelfand transforms with the Fourier transformsfa . . . . . f., this shows that the map q~: X ~ C " , ~0(X)-'=(fl(x) . . . . . f,(z)) (z~X) is injective and continuous on the compact Hausdorff space X, hence a homeomorphism onto ~o(X)c C". Consequently, X is metrizable and has finite dimension. ii)~iii) This is a consequence of (1) and FH-R] theorem (24.15). iii)~i) Suppose G is countable and has finite torsion-free rank n z N 0 . Our first claim is to prove the existence of functions f~ . . . . . f , + ~~ L ~(G) whose Fourier transforms separate the points of X: By definition of the torsion-free rank n of G (I-H-R], [Fu]), there exists an independent subset {Xl . . . . . x,} of the Z-module G consisting only of elements of infinite order and maximal with respect to these two properties. Let H be the subgroup of G generated by x~ . . . . . x, and let A(X, H) denote the annihilator {z~X: x(H)={1}} of H in X. Since G is countable, we may choose a countable set of representatives {gj: j z J } of the factor group G/H= { g j + H : jeJ}, where J = {1. . . . . m} for some m e n or J = I N . By the maximality of {x~, ..., x,}, G/H is a torsion group. This obviously implies that for every jEJ, {7(gj+H): 7 z ( G / H ) " } is a finite subgroup of the torus T. Identifying as usual (G/H) A with A(X, H), the latter observation proves the following fact which is crucial for the sequel:

VjeJ3ej>O: Iz(gj)-11 > ~j V~eA(X, H) s.th. x(gj) 4: 1,

(2)

Note that ~j only depends on gj. and that we may (and do) assume without loss of generality that ej < 1 holds for all j ~ J. j-1

Now put 5j.'=4 -~ I~ e, (/~J). Then we obviously have k=l

j-1 k=s+l

for every s, j ~ J such that j__>s + 1. For s EJ, set J,.'= }j 6 J: j >- s + 1} and observe that for every s~J we have

6j
~ j=s + 1

4"-J=3-16~es

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so that (Jj)j~s is a family of strictly positive real numbers such that J~-2

(3)

~ 6j>0 jedn

holds for every seJ. In the sequel, every other family of strictly positive real numbers satisfying (3) will serve as well as the one we constructed. Denoting by ~ the characteristic function of geG, we set

L,=l~

(l
jeJ

Then f t . . . . , f . + l obviously belong to La(G), and indeed fl ...... ~,+1 separate the points of X: Suppose we are given ct, f l e x such that fk(~)=fk(/~) for every 1 < k < n + l . Denoting complex conjugation by *, we have ~(Xk)=(fk(CO)*=(fk(~))*=fl(Xk) for every 1 <_k<_n+l, so that ~ coincides with fl on H, hence f l - ~ e A ( X , H). If ~ would be different from fl, there would exist a minimal s e J such that ~(g,) # fl(g~); we consequently would have

If. +~ ( ~ ) - L +~ (/3)1 = I,~ (~ (g,) - t~ tg~))* + Z aj(~ (g j) -/3(gj)3*l J~Js

j~Y~

>6~e~-2 ~ 6j>0, j~J~

since we may use (3) and (2) because fl-10~eA(X, H) satisfies fl-le(g.~)+ 1. But f,+ 1(~) + f , + l(fl) contradicts our initial assumption, hence e = fl, and our claim is proved. By (P), it is now clear that L 1(G) possesses a finite universal Korovkin system T. Explicitly, the set T..= 10,fl . . . . . f~+l, ~ fk*f~* (where * denotes the canonik=l

cal involution f * (g) = ( f ( - g))* of L 1(G)) consisting of n + 3 elements is a finite universal Korovkin system for L 1(G), see [AI 1, 2].

Remark I. The author gratefully acknowledges valuable correspondence with Prof. E. Kaniuth (Paderborn). The construction of the function f~ + 1 given above in the proof of the implication iii)~i) is modelled after his construction of an absolutely summable function f defined on a countable torsion group G and having an injective Fourier transform. Corollary 1. Let X be a compact metric abelian group of finite dimension n e N o. Then there exist 2 n + 2 continuous.functions q~k: X ~ JR whose Fourier transforms are absolutely summable on G = X and such that the map ~: X ~ J R 2"+2, ~b(Z) '=(~Pa (Z). . . . . cp2,+z(;D) is a homeomorphism onto its image.

Proof Set q~2k:=Ref~, ~02k-1 : = I m ~ for 1 < k < n + 1.

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Remark 2. Let X be a compact metric space of finite dimension n. Then by

a famous theorem of Monger and N6beling, X is homeomorphic to a subset of [0, l] z, +1, i.e. there exist 2 n + 1 continuous real-valued functions on X which separate points. A non-constructive proof of this theorem, which even yields the density of into-homeomorphisms in the space of all continuous, lR 2"+ ~-valued functions on X, may be found, together with some generalizations, in [H-W] or I-El.

Corollary 2. Let G be a discrete abelian group. Then the following assertions are equivalent: i) L 1(G) possesses a finite universal Korovkin system, ii) G is an extension of •" by a countable torsion group, iii) G is a countable extension of its torsion subgroup by a subgroup of Q", iv) G is an inductive limit of a countable number of groups of the form 7Z" x ~ , where each ~bj is a finite abelian group. Proof We just need to show the equivalence of ii)-iv) to the fact that G is

a countable group of finite torsion-free rank. Suppose the latter assertion is true. Then a subgroup H of G which is isomorphic to Z" and induces a countable torsion group as factor has been constructed in the proof of proposition 1. To see that iii) holds, observe that G and G/G~ (where G, denotes the torsion subgroup of G) have the same torsion-free rank and use the fact that the every torsion-free group of finite rank n is isomorphic to a subgroup of Q" ([Fu], [H-R]). Conversely, ii) as well as iii) obviously imply that G is countable, and G also has finite torsion-free rank, since in both cases this is true for a subgroup H as well as its corresponding factor G/H and we have to(G)= r0(H)+ ro(G/H) ([-Fu]). The proof is finished by the observation that iv) is equivalent to X = being an n-dimensional metric abelian group ([We] Sect. 29, p. 111).

Corollary 3. Let 0--* A ~ B ~ C ~ 0 be a short exact sequence of discrete abelian groups, i.e. B is an extension of A by C. Then L ~(B) possesses a finite universal Korovkin system, iff LI(A) as well as LI(C) do.

For the proof just note that B is countable and has finite torsion-free rank, iff A as well as C have: This follows again from the formula ro (B)= ro(A)+ ro (C). In particular, if B is a direct product of two abelian groups A, C, then L 1(A • C) possesses a finite universal Korovkin system iff LI(A) as well as LI(C) do. This is a special case of the fact that the projective tensor product A ~ B of two commutative unital Banach algebras possesses a finite universal Korovkin system, iff each factor does, see [Pa 5].

4. The General Case Using the structure theory of locally compact abelian groups and the techniques developed for the discrete case we may now give a characterization of those locally compact abelian groups which have property (P).

Theorem 1, Let G be a locally compact abelian group with character group X = G. Then the following assertions are equivalent: i) L 1(G) possesses a finite universal Korovkin system,

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M. Pannenberg ii) X is a finite-dimensional separable metric space.

Proof i ) ~ i i ) Let {fl . . . . . f,} be a finite universal Korovkin system for LtIG). If X o , = X ~ { 0 } is the one-point compactification of X and ~01. . . . . cp~ denote the continuous extensions off~ . . . . . f, to X o, we know that ~o~, ..., ~o, separate the points of the compact Hausdorff space X o by Theorem 4.1 of I-A13]. As shown in Proposition 1, this implies that X0 is homeomorphic to a subset of 112", hence X = X0\{0} is a separable metric space of finite dimension. i i ) ~ i ) Suppose X is a finite-dimensional separable metric space. Let us first assume that G possesses a compact open subgroup K the general case will be deduced from this later on. Since X has a countable base for its topology by ii), the same is true for G = ~ and its subgroup K. In particular, the dual g r o u p / ( is a countable discrete group, so that we may choose a sequence of characters Z,e (~ such t h a t / ( = {Z~IK: hEN}. Since K is open, the factor group G/K is a discrete group, whose dual group A(X, K) has finite dimension by ii). By proposition i), this implies that G/K is a countable discrete group of finite torsion-free rank r. Proceeding as in the proof of proposition 1, we may find elements x~, ..., x, eG such that the corresponding equivalence classes {xa + K . . . . . x, + K} form an independent set of elements of infinite order in G/K which is maximal with respect to these properties. Denoting the subgroup of G generated by K and x~ ..... Xr by H, we have that H is an open subgroup such that the corresponding discrete factor group G/H = (G/K)/(H/K) is a countable torsion group. Let { g / j ~ J } be a complete set of representatives of G/H = {gj + H: jeJ}. We are now ready to define the functions we are looking for in property (P): Denoting by 11~ the characteristic function of a subset S of G, we set

F0"= ~ 2-"Z,~K n=l

(g~G, 1 <=i<=r)

F1 (g):=Fo (g - x~)

F, +1 (g):= ~,, c~iFo (g -- g j)

(geO),

j~J

where (6i)j, J is an absolutely summable (possibly finite) sequence of strictly positive real numbers to the specified in the sequel. Observe that F0, F 1. . . . . F, are continuous functions with compact support, hence belong to LI(G), and that F , + ~ L ~(G) by the absolute summability of the family (6j)j~j. We claim that the above functions satisfy (P1) and (P2). To see that (P2) is fulfilled, take an arbitrary z ~ X and choose n ~ N such that xlx=xnlx. Since K is an open subgroup of G, the Haar measure of K is given by restricting the Haar measure of G to K; hence F o ( Z ) = f f ~ g , ) = 2 - " by the orthogonality relations for the compact group K. This shows Fo(Z)>0 for every x e X and may also used to prove (P1): Let ~, f l e x be given such that ~(cQ=Pi(fl) for every 0 < i < r + l , By what we have just seen, i0o(~)=/eo(fl) implies ~IK=/~IK. For every 1 < i < r , we have

(x ,)),

--

Ix,))* ,eo I8)

(x ,)), Fo

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457

so that o~(x~)=fl(xi) because ffo(~)>0. Summing up, this shows that e coincides with fl on the subgroup H, i.e. fl- 1c~eA(X, H). If c~ were different from /3, there would exist a minimal seJ such that c~(g~) 4=/~(g,). Then we would have

IF, +~ (~)-

F, +, (/~)1 =

I Z ,sA~(gs)- [~(gg)* Fo(~)1 ./eJ

_->~eolCe)(6~I(;-' ~)(g~)- 11- 2 y 6j), J~-/s

which is strictly positive if (6s)s~s is appropriately chosen as in the proof of Proposition 1 (note that we may use a relation like (2) since again G/H is a torsion group, //-lct~A(X, H) and (1~ lc0(g~)=#l ). This contradiction to our initial assumption shows that we must have a=/~ and finishes the proof for those groups G which possess a compact open subgroup. We note in passing that the number of functions constructed above may be related to the dimension of X = G: Since dim X = r, we need at most dim X + 2 functions to fulfill (P). In the general case, we may use a well-known theorem of harmonic analysis ([H-R], [Well to decompose the locally compact abelian group G in the form G = I R " x Go, where n e N o is a uniquely determined nonnegative integer and Go is a locally compact abelian group containing a compact open subgroup. Since Go is topologically isomorphic to A(X, lR~x {0}), we know that Go is a separable metric space of finite dimension, and may thus apply what we have already proved to find as above functions F0, F~, ..., FsELI(G) whose Fourier transforms separate the points of Go. As shown above, we may even assume /e0(~) > 0 for every ~eGo. Now choose functions fo . . . . . f, e L ~(IR") such that fo(t) = e x p ( - ] l t / ] 2),

fk(t)=tkexp(--!]tll 2)

(l_--
holds for every t=(tl ..... t,)EN ~, where rl II denotes the euclidean norm. Note that f0 > 0 and set

Cbk,=fk| o Hi :=fo | Fi

(O
where we use the standard notation ( f | F)(x, g ) = f ( x ) F(g). Then the finite set of functions T={@o . . . . . 4~,, H1 . . . . . H~} is obviously contained in LI(G) and has the desired properties (P1) and (P2): This follows from the relation ( f | F) ^ = f | P and an elementary calculation that is left to the reader.

Remark 3. The Menger-N6beling theorem, as mentioned in Remark 2, also holds for arbitrary (not necessarily compact) separable metric spaces, see [H-W]. Using the following lemma, we may relate the number of Fourier transforms construted above to the dimension of the finite-dimensional separable metric space X and thus get a version of the Menger-N6beling theroem for the particular case of a second countable LCA group X of finite dimension d which allows to imbed

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X into IR 2a+3 (instead of N2a+ 1) by a homeomorphism whose coordinate functions are Fourier transforms. Lemma 1. Let X1, X2 be two separable metric LCA groups, and let X o = X 1 x X2 denote their direct product. Then Xo is a separable metric LCA group of dimension dim Xo = dim X ~+ dim X2. To prove the lemma, one may use the structure theory of LCA groups and a theorem of Hurewicz ([Hu]) to reduce the proof to the case of a compact group and note that for compact groups, the assertion of the lemma is an immediate consequence of Pontryagin's theorem and the equality to(X1 x X2) =r0(R1)+ro(~2). Alternatively, the lemma follows from the equality dim G = d i m H + d i m G/H valid for every locally compact group G and any of its closed subgroups H - the proof of this formula is not obvious at all and to be found in [Y] for the separable metric and in [Na] for the general case. It should be mentioned that the logarithmic formula dim (X x Y) = dim X + dim Y does not hold in general ([H-W], p. 33-34), not even for topological groups ([K]). Now let G = 1R" x Go be a decomposition as in the proof of Theorem 1. Then X=~=~"x(~0, so that d i m X = n + d i m ( ~ o by the above lemma. But since dim (~o + 2 = s § 1, we may find by the constructions of Theorem 1 n + s § I = dim X + 2 complex-valued functions in L I(G) with (P 1) and (P 2). Since 43o > 0, this yields a homeomorphism of X into R 2~/+3 (where d--dim X) whose coordinate functions are Fourier transforms of real-valued functions which are absolutely integrable on G = ~. To obtain a dual characterization of assertion ii) in Theorem 1, we give the following proposition, which is a combination of a theorem of Pontryagin and an observation of Weil and characterizes ii) by the finiteness of the torsionfree rank of a certain discrete factor of G. For a compact subset N of G with strictly positive Haar measure and a real number 0 < e < 1 we use the notation P(N, e),={z~X: I z ( x ) - 11< e for all x~N}. The closed subgroup of compact elements of G ([H-R]) will be denoted by G c.

Proposition 2. Let G be a locally compact abelian group, which is a separable metric space. For every compact subset N of G of strictly positive Haar measure and every real number 0 < e < 1 we denote the closed subgroup of G generated by the connected component C of the identity in G and the annihilator A(G, P(N, e)) by Hu.,. Then the following assertions are valid: 1) Hm~ is an open and closed subgroup of G. 2) The torsion-free rank of G/HN, ~ is finite, iff the dimension of X = G is finite.

Proof. U.'=P(N, e) is an open symmetric neighbourhood of the identity in X, which has compact closure by our assumptions on N and e ([H-R], Cor. 23.16). Let F be the subgroup of X algebraically generated by U. Then F is a clopen subgroup of X, hence a locally compact subgroup in its own right which by definition is compactly generated. By the structure theorem for such groups, we may decompose F in the form F = P , " x ZSx K, where n, s are nonnegative

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integers and K is a compact abelian group. Now by our assumption on G, X is second countable, dim X = d i m F ([We], Sect. 29) and dim(N n x T/Sx K) = n + d i m K by Lemma I, so that we have d i m X < o v - c ~ d i m K < o o r has finite torsion-free rank, where the last equivalence is a consequence of Pontryagins result (l). We further determine K: Since K = F c = X C n F , we have that A(G, K ) = A ( G , XCc~F) is the closed subgroup of G generated by A(G, X c) and A(G, F). Since A(G, X')= C and obviously A(G, F ) = A ( G , P(N, e~))this shows A(G, K)= H~,~. Consequently, HN, ~ is open as annihilator of the compact group K (which proves 1)) and we have dim X < ~ ~G/HN.~ = G/A(G, K ) = / ~ has finite torsion-free rank, which finaIly proves 2). Using the above notations, this yields

Corollary 4. Let G be a locally compact abelian group. Then the following assertions are equivalent: i) L 1(G) possesses a finite universal Korovkin system ii) G is separable metric and there exists a compact subset N of G of strictly positive Haar measure and a real number 0 < ~ < 1 such that G/H~,,~ has finite torsion-free rank. The proof is an immediate consequence of the preceding proposition and the fact that G is second countable iff X = G is (l-H-R], [Po]). Denoting the Fourier algebra of X = d by d ( X ) = { f : f e L l ( G ) } , we may now prove the following characterization of finite-dimensional separable metric abelian groups:

Theorem 2. Let X be a locally compact abelian group. Then the following assertions are equivalent: i) There exists a finite number of functions fl . . . . . f, e d ( X ) which separate the points of X and do not all vanish in one point of X ii) X is a finite-dimensional separable metric group. Proof. The equivalence of i) and ii) follows from Theorem 1 and the fact that G = I~ has (P) iff i) is true.

Corollary 5. Let X be a second countable LCA group. Then the following assertions are equivalent: i) There exists a finite number of functions f l , . . . , f ~ e d ( X ) which separate the points of X and do not all vanish in one point of X ii) X is the local product of an abetian n-parameter local Lie group and an abelian compact totally disconnected group. Proof. The assertions i) and ii) are equivalent by the preceding theorem and a famous theorem of Montgomery and Zippin ([M-Z], Theorem 4.9.3, p. 184).

5. Comments and Corollaries In the absence of non-trivial compact subgroups of G we may dispose of the second countability assumption:

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Corollary 6. Let G be a locally compact abelian group containing no non-trivial compact subgroups. Then L 1(G) possesses a finite universal Korovkin system iff X = G has finite homological (= cohomological) dimension. Proof. Since G contains no non-trivial compact subgroups, its character group

X is a connected LCA group. If X has also finite homological dimension, it is a separable metric ([Ne]) LCA group of finite dimension, hence L 1(G) possesses a finite universal Korovkin system by Theorem 1. The converse is a direct consequence of Theorem 1 and the coincidence of covering dimension and homological dimension for separable metric spaces. Theorem l may be used to show that an LCA group G has property (P) iff each of its closed subgroups as well as each of its Hausdorff factor groups has property (P~. Corollary 7. Let G be a locally compact abelian group, and let H ~ G be a closed subgroup. Then G has property (P) iff H as well as G/H have property (P). Proof By Theorem 1 and a standard duality argument, it is sufficient to prove

that an LCA group X is a separable metric space of finite dimension, iff an arbitrary closed subgroup as well as the corresponding factor group is such a space. Let X be an LCA group and Y a closed subgroup. Then X is a separable metric space iff Y as well as X/Y are: One half of this assertion is easy to see, the other implication may be proved following the arguments given in Theorem 5.38 (e), (fl of [H-R]. An analogous assertion holds for the dimension: Let X be a separable metric LCA group and u a closed subgroup. Then X is finite-dimensional iff Y as well as X/Y are. This result is by no means trivial (compare the errata section of Well's book ([-We], p.158)); it may be proved using a result of Deane Montgomery ([Mo]) for one implication and a theorem of Hurewicz ([H-W], Theorem VI.7; compare also [Le] Theorem 8) for the other implication. Alternatively, it is a direct consequence of the formula dim X = dim Y + dim (X/Y) (l-Y], [Na]). The analogue of Corollary 3 also holds in the general case, if one considers only strict exact sequences, i.e. exact sequences where the homomorphisms involved are continuous maps which are also open onto their image: p

Corollary 8. Let 0 ~ A

,B , C -o 0 be a short strict exact sequence of locally compact abelian groups. Then L 1(B) possesses a finite universal Korovkin system, /ff LI(A) as well as LI(C) do.

That the sequence is strict exact just means that A ~ H . . = i ( A ) and C ~ B / H are topological isomorphisms; consequently the above equivalence is a direct consequence of the preceding corollary. To deduce a version of Theorem 1 for generalized group algebras, let B be a commutative Banach algebra with an isometric symmetric involution and a bounded approximate identity. If G is a locally compact abelian group, we denote by LI(G, B) the generalized group algebra induced by the absolutely integrable B-valued functions on G ([Ma] XI.5). Then LI(G, B ) = L 1 (G)~,~B is the projective tensor product of the involutive Banaeh algebras LI(G) and B; hence L I(G, B) is a commutative Banaeh algebra with an isometric symmetric

Characterization of LCA Groups via Korovkin Theory

461

involution and a bounded approximate identity. For this generalized group algebra we get the following generalization of Theorem 1 : Theorem 3. Let G be a locally abefian group with character group X = ~ and B a commutative Banach algebra with an isometric symmetric involution and a bounded approximate identity. Then the following assertions are equivalent: i) L I (G, B) possesses a finite universal Korovkin system, ii) X is a finite-dimensional separclbIe metric space and B possesses a finite universal Korovkin system. Using the tensor product representation L 1 (G, B)= L I(G) ~ , B , this follows immediately from the subsequent proposition showing that the existence of a finite universal Korovkin system is a hereditary as well as a permanent property with respect to the formation of projective tensor products. The properties of the projective tensor product of Banach algebras used in the sequel are to be found e.g. in [Ma].

Proposition 3. Let

A, B be commutative Banach algebras with isometric symmetric involution and a bounded approximate identity. Then A Q , B possesses a finite universal Korovkin system iff A as well as B do.

Proof. Under the above hypotheses, the existence of a finite universal Korovkin system in each of the algebras A, B, A ~ B is equivalent to the existence of a finite number of elements in the algebra whose Gelfand transforms strongly separate the points of the maximal ideal spaces AA, AB, AAQ~B_~A A • AB; compare Theorem 4.1 and Corollary 4.5 of [A13]. Suppose T~ A ~ B is a finite subset such that T.'={~: z~ T} (where A denotes the Gelfand transformation) strongly separates the points of A A x A~. For arbitrary but fixed mlEAa, mzEAB and each z ~ A ~ B we denote by zm,~B resp. zm2eA arbitrary elements of the respective algebra whose Gelfand transforms satisfy 2,,,(fl)=~(ml, /3) resp. 2m2(ZQ=~[C~, m2) for all fleA B resp. o:~AA; these are uniquely determined only if A resp. B are semisimple. Now set Tt..={z,,~: z ~ T } resp. T2:={z~: z e T } . Then both sets are finite and the corresponding Gelfand transforms obviously separate the points of AB resp. A A . Conversely, suppose the Gelfand transforms of the elements x~ . . . . . x, e A resp. Yt, -.-, y~eB strongly separate the points of A A resp. AR. Then set Zo,j,k:=Xj(~yk

and

Zi,L~:=XiXj(~y k

(l<=i,j
and consider T.'----{zi.j.k:O<__i<=r, l <j<=r, 1 <=k<=s}. Then Tis a finite subset of A | whose Gelfand transforms strongly separate the points of LIA(~B~--AA • A~: This may be seen by a straightforward computation whose proof is left to the reader. In the setting of commutative unital Banach algebras, where no involution is at hand, the existence of a finite universal Korovkin system is no longer equivalent to the existence of a finite subset whose Gelfand transforms separate points. Nevertheless, analogues of the above proposition are true and may be found in [Pa5]. Remark 4. The characterization of those locally compact abelian groups G for which L ~(G) possesses a countable universal Korovkin system is nearly trivial:

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This is the case exactly for the separable metric groups. If G is a separable metric group, its ~r-algebra of Borel sets is countably generated, hence the group algebra Lt(G) is separable and so obviously possesses a countable universal Korovkin system. Conversely, if L 1(G) possesses a countable universal Korovkin system, it follows from Theorem 4.1 of [A13] that there exists on X=(~ a countable set of continuous complex-valued functions vanishing at infinity and satisfying (P1), (P2), which shows that the dual group X = G is homeomorphic to a subset of the separable metric space IE~, hence a separable metric space in its own right, so that the same is true for G. Remark 5. A general characterization of those commutative unital Banach alge-

bras possessing a finite universal Korovkin system is not known (see, however, [Pa 3]). Yet, if one considers Korovkin approximation of lattice homomorphisms by positive linear operators defined on a Banach lattice E, one has the following characterization due to Wolff [Wo]: A Banach lattice E possesses a finite universal Korovkin system~=~E is finitely generated. The verbatim analogue remains true for particular C*-algebras ([Be]), but fails for commutative unital Banach algebras ([Pa2]). If E = L~(G) denotes the Banach lattice of all absolutely Haar-integrable real-valued functions on a locally compact abelian group G, then E has order-continuous norm, and Corollary 2.10 of [Wo] allows the following characterization of those LCA groups G for which the Banach lattice E = L ~ ( G ) possesses a finite universal Korovkin system in the sense of ]-Wo], i.e. with respect to the approximation in L 1-norm of lattice hemomorphisms taking their values in an arbitrary Banach lattice by equicontinuous sequences of positive operators on E: L~t(G) possesses a finite universal Korovkin system,~ G is a separable metric group. Indeed, the first assertion is equivalent to E = L~(G) being separable by Corollary 2.10 of [Wo]. Since L~(G) is separable, iff G is a separable metric group, the above equivalence is obvious. It is shown in [Wo] that E even possesses a three-element universal Korovkin system in case G is separable metric. The above equivalence clearly shows the differences between Korovkin approximation in the Banach lattice L~(G) ([B-L], [D 1, 2], [Bri], [K-W], [Lo]) resp. the Banach algebra L~(G). One reason for this difference is the fact that we are considering convergence in the Ll-norm resp. in the uniform norm, compare [Pa2], Cor. 5.9. Remark 6. It seems interesting to investigate if the characterization of finitedimensional second countable locally compact abelian groups established in this note also holds for a general not necessarily abelian locally compact group G in the following form:

G is second countable and has finite dimension,~,There exists a finite number of functions in d ( G ) separating strongly the points of G (=, G has property (P))

(4)

Here ~/(G) denotes the Fourier algebra of G as introduced by Eymard in [Ey]. The other implication being clear, we have to decide if the finite dimension of a second countable locally compact group G implies G has (P).

Characterization of LCA Groups via Korovkin Theory

463

It m a y be proved that a second c o u n t a b l e totally d i s c o n n e c t e d g r o u p has (P): F o l l o w i n g a n idea of E. K a n i u t h , one m a y construct a strictly positive injective f u n c t i o n in ~ ' ( G ) using a n absolutely c o n v e r g e n t series of a p p r o p r i a t e multiples of translates of characteristic functions of c o m p a c t o p e n s u b g r o u p s in G. Second c o u n t a b l e Lie groups also have (P): This follows from W h i t n e y ' s e m b e d d i n g t h e o r e m for differential manifolds a n d P r o p o s i t i o n 3.26 of [Ey]. U s i n g T h e o r e m 4.10.1 of [ M - Z l one n o w sees that (4) holds at least for every locally c o n n e c t e d locally c o m p a c t group. I m i t a t i n g the last step in the p r o o f of T h e o r e m 1 one m a y show that (P) carries over to a direct p r o d u c t of groups. So at least finite products of second c o u n t a b l e Lie groups a n d second c o u n t a b l e totally disconnected groups have property (P).

Acknowledgement. The author would like to thank Professor E. Kaniuth for valuable correspondence on the subject of this paper.

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Received May 13, 1988; in final form September 27, 1989