Induction for Banach algebras, groupoids and KK^ban

TL;DR

This paper proves that the Bost conjecture holds for equivalent groupoids with Banach algebra or C*-coefficients, using functoriality of KK-theory and Morita equivalence of associated algebras.

Contribution

It establishes the functoriality of Lafforgue's KK-theory for groupoids and Banach algebras and demonstrates Morita equivalence of L^1-algebras for equivalent groupoids.

Findings

Bost conjecture invariance under groupoid equivalence

Functoriality of KK-theory for Banach algebras and groupoids

Morita equivalence of L^1-algebras for equivalent groupoids

Abstract

Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C*-coefficients. To show these results, the functoriality of Lafforgue's KK-theory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L^1-algebras (with Banach algebra coefficients).