The spectral radius in $\mathcal{C}_0(X)$-Banach algebras

Abstract

The spectral radius of an element of a C 0 .X /-Banach algebra can be calculated as the supremum of the spectral radii in the fibres. As a consequence, calculations in K-theory for such algebras can potentially be carried out fibrewise.

The Bost conjecture and proper Banach algebras

“…Proceed inductively to define the assembly map for all n 2 N 0 . Note that à B is an isomorphism in K-theory by the results of [Par13a].…”

“…Note that diagram (2) gives the result directly only for D 0, but we can apply it to S n B instead of B and use the invariance results for K-theory of [Par13a] to obtain it for arbitrary degrees. This applies in particular to the regular unconditional completion L 1 .G / and its symmetrised version…”