Commutativity of the Haagerup tensor product and base change for operator modules

TL;DR

This paper characterizes the Beck-Chevalley condition for base change of operator modules over commutative C*-algebras using the Haagerup tensor product, and proves a descent theorem for continuous fields of Hilbert spaces.

Contribution

It provides a new characterization of the Beck-Chevalley condition via the completely bounded norm of the flip map on the Haagerup tensor product.

Findings

Characterization of the Beck-Chevalley condition for operator modules

A descent theorem for continuous fields of Hilbert spaces

Analysis of the completely bounded norm of the flip map

Abstract

By computing the completely bounded norm of the flip map on the Haagerup tensor product $C_0 Y_1\otimes_{C_0 X} C_0 Y_2$ associated to a pair of continuous mappings of locally compact Hausdorff spaces $Y_1\rightarrow X\leftarrow Y_2$, we establish a simple characterisation of the Beck-Chevalley condition for base change of operator modules over commutative $C^*$-algebras, and a descent theorem for continuous fields of Hilbert spaces.