A generalised Green-Julg theorem for proper groupoids and Banach algebras

TL;DR

This paper generalizes the Green-Julg theorem from compact groups to proper groupoids and Banach algebras, establishing new connections in K-theory and the Bost assembly map.

Contribution

It extends the Green-Julg theorem to proper groupoids and Banach algebras, and shows the surjectivity of the Bost assembly map for proper Banach algebras.

Findings

Green-Julg theorem generalized to proper groupoids

Bost assembly map shown to be surjective for proper Banach algebras

Spectral radius in C_0(X)-Banach algebras can be computed fiberwise

Abstract

The Green-Julg theorem states that K_0^G(B) is isomorphic to K_0(L^1(G,B)) for every compact group G and every G-C*-algebra B. We formulate a generalisation of this result to proper groupoids and Banach algebras and deduce that the Bost assembly map is surjective for proper Banach algebras. On the way, we show that the spectral radius of an element in a C_0(X)-Banach algebra can be calculated from the spectral radius in the fibres.