[Paper Review] Tensor products with bounded continuous functions

Dana P. Williams|ArXiv.org|Jul 9, 2003

Advanced Topology and Set Theory8 references3 citations

TL;DR

This paper investigates tensor product isomorphisms between spaces of bounded continuous functions and C*-algebras, showing that $ C^b(X) \otimes A \to C^b(X,A) $ is an isomorphism if and only if $ X $ is pseudocompact. It further proves that $ C^b(X,C^b(Y)) \to C^b(X\times Y) $ is an isomorphism precisely when both $ X $ and $ Y $ are pseudocompact, leading to a characterization of when the Stone-Čech compactification of a product space is the product of the compactifications.

ABSTRACT

We study that natural inclusions $C^b(X) ensor A$ into $C^b(X,A)$ and $C^b(X, C^b(Y))$ into $C^b(X imes Y)$. In particular, excepting trivial cases, both these maps are isomorphisms only when $X$ and $Y$ are pseudocompact. This implies a result of Glicksberg showing that the Stone-Cech compactificiation $β(X imes Y)$ is naturally identified with $βX imes βY$ if and only if $X$ and $Y$ are pseudocompact.

Motivation & Objective

• To determine when the natural map $ C^b(X) \otimes A \to C^b(X,A) $ is an isomorphism for a locally compact Hausdorff space $ X $ and a C*-algebra $ A $.
• To analyze the isomorphism condition for the map $ C^b(X,C^b(Y)) \to C^b(X\times Y) $, particularly under the norm topology.
• To establish a topological characterization of pseudocompactness via tensor product isomorphisms.
• To recover and reprove Glicksberg’s result on the product structure of Stone-Čech compactifications using tensor product techniques.

Proposed method

• Uses the supremum norm on $ C^b(X,A) $ to define a C*-norm on the algebraic tensor product $ C^b(X) \odot A $, leading to the completion $ C^b(X) \otimes A $.
• Proves that $ f \in C^b(X,A) $ lies in $ C^b(X) \otimes A $ if and only if the range of $ f $ has compact closure.
• Applies partition of unity arguments under the assumption of local compactness and paracompactness to construct finite covers and approximate functions in the tensor product.
• Employs Frolík’s Lemma to show that the supremum norm of differences $ \|f(x) - f(x_0)\|_\infty $ is continuous when $ Y $ is pseudocompact.
• Uses a contradiction argument with nested neighborhoods and sequences to show that if $ Y $ is not pseudocompact, the map $ \iota_2 $ fails to be surjective.
• Leverages pseudocompactness of $ X \times Y $ to extract convergent subsequences in $ C^b(Y) $, leading to a contradiction when $ Y $ is not pseudocompact.

Experimental results

Research questions

• RQ1 Under what conditions is the natural map $ \iota_1: C^b(X) \otimes A \to C^b(X,A) $ an isomorphism for a C*-algebra $ A $?
• RQ2 When is the map $ \iota_2: C^b(X,C^b(Y)) \to C^b(X\times Y) $ an isomorphism under the norm topology?
• RQ3 How does pseudocompactness of $ X $ and $ Y $ affect the structure of the Stone-Čech compactification of $ X \times Y $?
• RQ4 What is the relationship between the range of a function in $ C^b(X,A) $ and its membership in the completed tensor product $ C^b(X) \otimes A $?
• RQ5 Can Glicksberg’s result on $ \beta(X \times Y) \cong \beta X \times \beta Y $ be derived from tensor product isomorphisms?

Key findings

• The map $ \iota_1: C^b(X) \otimes A \to C^b(X,A) $ is an isomorphism if and only if $ X $ is pseudocompact, provided $ A $ is infinite-dimensional.
• The map $ \iota_2: C^b(X,C^b(Y)) \to C^b(X\times Y) $ is an isomorphism if and only if $ Y $ is pseudocompact, assuming $ X $ is infinite and pseudocompact.
• If $ X $ and $ Y $ are pseudocompact, then $ \beta(X \times Y) \cong \beta X \times \beta Y $ via the natural extension of the product map.
• The range of a function $ f \in C^b(X,A) $ is precompact if and only if $ f \in C^b(X) \otimes A $, which characterizes the image of the tensor product map.
• When $ Y $ is not pseudocompact, a counterexample function $ F \in C^b(X\times Y) $ can be constructed such that the associated $ f(x) = F(x,\cdot) $ is not continuous, showing $ \iota_2 $ is not surjective.
• The proof relies on constructing a sequence $ \{x_n\} $ in $ X $ and disjoint precompact open sets $ \{V_n\} $ in $ Y $, leading to a contradiction when assuming continuity of $ f $ under pseudocompactness.