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

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 $Y_1\to X\leftarrow Y_2$, we establish a simple characterisation of the Beck-Chevalley condition for base change of operator modules over commutative $C^{\ast }$-algebras, and a descent theorem for continuous fields of Hilbert spaces.

10 pages. Minor changes to the exposition; corrected an error in Remarks 6 not affecting the rest of the paper