Tensor products with bounded continuous functions

TL;DR

This paper investigates the conditions under which natural tensor product inclusions of bounded continuous functions are isomorphisms, revealing that pseudocompactness of spaces is key to these identifications and related compactification results.

Contribution

It establishes that these tensor product maps are isomorphisms only when the spaces are pseudocompact, clarifying the structure of bounded continuous function spaces and their relation to compactifications.

Findings

Tensor product inclusions are isomorphisms only for pseudocompact spaces.

Identifies when the Stone-Cech compactification of a product equals the product of compactifications.

Provides conditions linking function space tensor products to topological properties of spaces.

Abstract

We study that natural inclusions $C^b(X) \tensor A$ into $C^b(X,A)$ and $C^b(X, C^b(Y))$ into $C^b(X \times Y)$. 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 $\beta(X \times Y)$ is naturally identified with $\beta X \times \beta Y$ if and only if $X$ and $Y$ are pseudocompact.