Tensor products of C(X)-algebras over C(X)

TL;DR

This paper investigates the structure of tensor products of C(X)-algebras over C(X), focusing on the existence of minimal and maximal C^*-norms and the conditions under which C^*-norms exist.

Contribution

It characterizes the conditions for the existence of C^*-norms on tensor products of C(X)-algebras, especially when one algebra defines a continuous field over X.

Findings

Existence of minimal and maximal C^*-norms when one algebra is a continuous field.

No general C^*-norm exists on the tensor product without additional conditions.

Abstract

Given a Hausdorff compact space X, we study the C^*-(semi)-norms on the algebraic tensor product $A\otimes_{alg,C(X)} B$ of two C(X)-algebras A and B over C(X). In particular, if one of the two C(X)-algebras defines a continuous field of C^*-algebras over X, there exist minimal and maximal C^*-norms on $A\otimes_{alg,C(X)} B$ but there does not exist any C^*-norm on $A\otimes_{alg,C(X)} B$ in general.