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Tensor Products with Bounded Continuous Functions New York Journal of Mathematics New York J. Math. 9 (2003) 69–77. Tensor products with bounded continuous functions Dana P. Williams b b Abstract. We study the natural inclusions of C (X) ⊗ A into C (X, A) and Cb X, Cb(Y ) into Cb(X × Y ). In particular, excepting trivial cases, both these maps are isomorphisms only when X and Y are pseudocompact. This implies a result ofGlicksberg showing that the Stone- Cechˇ compactificiation β(X × Y ) is naturally identified with βX × βY ifand only if X and Y are pseudocompact. Contents 1. Main results 70 2. Glicksberg’s theorem 72 3. Pseudocompact spaces 74 4. Proof of Theorem 3 75 References 77 Suppose that X is a locally compact Hausdorff space and that A is a C∗-algebra. The collection Cb(X, A) of bounded continuous A-valued functions on X is a C∗- algebra with respect to the supremum norm. (When A = C, we write simply Cb(X)). Elements in the algebraic tensor product Cb(X)A will always be viewed as functions in Cb(X, A), and the supremum norm on Cb(X, A) restricts to a C∗- b b norm on C (X) A. Thus we obtain an injection ι1 of the completion C (X) ⊗ A into Cb(X, A): b b ι1 : C (X) ⊗ A→ C (X, A), and we can identify Cb(X) ⊗ A with a subalgebra of Cb(X, A). It is one of the fundamental examples in the theory that ι1 is an isomorphism in the case that X is compact [8, Proposition B.16], and we want to investigate the general case here. b Our main result identifies the range of ι1 as those functions in C (X, A) whose range has compact closure. As a consequence we show — provided A is infinite di- mensional — that ι1 is an isomorphism if and only if X has the property that every Received April 14, 2003. Mathematics Subject Classification. Primary: 46L06; Secondary: 54D35, 54D20. Key words and phrases. Tensor products, C∗-algebras, Stone-Cechˇ compactification, pseudo- compact. ISSN 1076-9803/03 69 70 Dana P. Williams continuous function on X is bounded. Such spaces are called pseudocompact. Since paracompact pseudocompact spaces are compact, it was tempting to make a blan- ket assumption of paracompactness. However, the arguments here use properties naturally associated to pseudocompactness rather than compactness, so it seemed worth the little bit of extra effort to include the general results. (Nevertheless, I have tried to organize the paper so that the niceties of noncompact pseudocompact spaces come at the end.) In fact, pseudocompactness arises again when we consider the case where A = Cb(Y ) for a locally compact Hausdorff space Y . Then we are led to address the properties of the natural inclusion b b b ι2 : C X, C (Y ) → C (X × Y ). In general, this map is an isomorphism when Cb(Y ) is given the strict topology viewed as the multiplier algebra of C0(Y )[1, Corollary 3.4]. Here, however, we are interested in the norm topology, and in Theorem 3 we show that ι2 is an isomorphism if and — assuming that X is both infinite and pseudocompact — only if Y is pseudocompact. Combining these observations about ι1 and ι2 yields a well-known result about products of Stone-Cechˇ compactifications originally due to Glicksberg [5, Theorem 1] with simplified proofs given by Frol´ık [4] and Todd [10]. Recall that if X is a locally compact, then, following [3, §XI.8.2] for example, the Stone-Cechˇ compactification of X is a compact Hausdorff space βX together with a homeomorphism iX of X onto a dense open subset of βX so that (βX,iX ) has the extension property: given any continuous map f of X into a compact Hausdorff space Y , there is a continuous map f ∗ : βX → Y such that f = f ∗ ◦ iX . (The pair (βX,iX ) is unique up to the natural notion of equivalence.) In particular, if Y is locally compact Hausdorff, the extension property gives a surjection ϕ : β(X×Y ) → βX × βY . Our results can be used to show that ϕ is a homeomorphism if and only if X and Y are pseudocompact (which is the special case of Glicksberg’s result alluded to above). 1. Main results Recall that a subset of a topological space is called precompact if its closure is compact. Theorem 1. If X is a locally compact Hausdorff space and if A is a C∗-algebra, then f ∈ Cb(X, A) is in Cb(X) ⊗ A if and only if the range of f, R(f):={f(x): x ∈ X}, is precompact. Proof. Suppose that f ∈ Cb(X) ⊗ A. To show that R(f) is precompact, it will suffice to see that R(f) is totally bounded. So we fix>0 and try to show that n b R(f) can be covered by finitely many -balls. Choose i=1 fi ⊗ ai ∈ C (X) A so that n f − f ⊗ a < . i i 2 i=1 ∞ Since n n n f ⊗ a (x) − f ⊗ a (y) ≤ f (x) − f (y) a , i i i i i i i i=1 i=1 i=1 Tensor products with bounded continuous functions 71 b and since the image of each f ∈ C (X) is bounded, we can find points x1,...,xr ∈ X such that the open sets n n U := x ∈ X : f ⊗ a (x) − f ⊗ a (x ) < k i i i i k 2 i=1 i=1 n cover X. For convenience, let bk := i=1 fi ⊗ ai (xk), and let B(bk)={a ∈ A : a − bk <}, be the ball of radius at b .Nowifx ∈ U , then k k n n n f(x) − f ⊗ a (x ) ≤ f(x) − f ⊗ a (x) + f ⊗ a (x) − b i i k i i i i k i=1 i=1 i=1 < + = . 2 2 Since the Uk cover X, it follows that r R(f) ⊂ B(bk). k=1 Thus R(f) is totally bounded, and R(f) must be compact. Now assume that R(f) is compact and therefore totally bounded. Thus given r >0, there are elements {bk}k=1 ⊂ A such that r R(f) ⊂ B(bk). k=1 −1 r Let Uk := f B(bk) . Then U = {Uk}k=1 is an open cover of X. First, we suppose that there is a partition of unity on X subordinate to U. That is, we b assume that there are functions fk ∈C (X) with 0 ≤ fk ≤ 1, supp fk ⊂ Uk and r r b k=1 fk(x) = 1 for all x ∈ X. Then k=1 fk ⊗ bk ∈ C (X) A, and for all x ∈ X, we have r r r f ⊗ b (x) − f(x) = f (x)b − f (x)f(x) k k k k k k=1 k=1 k=1 n ≤ fk(x) bk − f(x) k=1 n ≤ fk(x)=. k=1 Thus, if such a partition of unity exists, then f belongs to the closure of Cb(X) A and f ∈ Cb(X)⊗A as required. If X were paracompact, then given any finite cover r U, there is a partition of unity {fk}k=1 on X subordinate to U [8, Proposition 4.34]. Since, out of stubbornness, X is not assumed to be paracompact, we will have to take advantage of the special nature of the covers U involved and the extension property of the Stone-Cechˇ compactification βX of X. Since R(f) is compact and f : X → R(f) is continuous, there is a continuous function F : βX → R(f) such X −1 that F ◦ i = f. Let Vk := F B(bk) . Then V = {Vk} is a finite open cover of βX, and there is a partition of unity {ϕk} on βX subordinate to V. Since 72 Dana P. Williams X −1 X Uk =(i ) (Vk), it follows that if fk := ϕk ◦ i , then {fk} is a partition of unity on X subordinate to U. This completes the proof. Corollary 2. Suppose that X is a locally compact Hausdorff space and that A is a C∗-algebra. If X is pseudocompact, then R(f) is compact for each f ∈ Cb(X, A), and (1) Cb(X) ⊗ A = Cb(X, A) (that is, ι1 is an isomorphism). Conversely, if A is not finite dimensional, then (1) holds only if X is pseudocompact. If A is finite dimensional, then (1) always holds. Proof. Suppose that X is pseudocompact. It is not hard to see that the continuous image of a paracompact space in a metric space is compact (Corollary 8). Thus R(f) is compact for each f ∈ Cb(X, A) and (1) follows from Theorem 1. Now assume that A is infinite dimensional and that X is not pseudocompact. It follows from [9, Theorem 1.23], that the unit ball of A is not compact. Thus there ∞ ∞ is a sequence {ak}k=1 of elements of A of norm at most one such that {ak}k=1 is not totally bounded. Since X is not pseudocompact, there are precompact open sets Un in X whose closures are locally finite and pairwise disjoint (Lemma 6). Fix xn ∈ Un. Then by Urysohn’s Lemma (cf. [7, Proposition 1.7.5]), there are fn ∈ Cc(X) such that 0 ≤ fn ≤ 1, fn(xn) = 1 and supp fn ⊂ Un. Since {Un} is locally finite, ∞ f(x):= fn(x)an n=1 b ∞ defines a function f ∈ C (X, A) whose range contains {an}n=1.Thusf is not in the range of ι1, and (1) does not hold.
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