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

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.