Spaces of sections of Banach algebra bundles

Abstract

Suppose that $B$ is a $G$-Banach algebra over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, $X$ is a finite dimensional compact metric space, $\zeta :P\to X$ is a standard principal $G$-bundle, and $A_{\zeta }=\Gamma (X,P{\times }_{G}B)$ is the associated algebra of sections. We produce a spectral sequence which converges to ${\pi }_{\ast }(G{L}_o{A}_{\zeta })$ with [E^2_{-p,q} \cong \check{H}^p(X ; π_q(GL_o B)).] A related spectral sequence converging to $\K_{*+1}(A_ζ)$ (the real or complex topological $K$-theory) allows us to conclude that if $B$ is Bott-stable, (i.e., if $ π_*(GL_o B) \to \K_{*+1}(B)$ is an isomorphism for all $\ast >0$) then so is $A_{\zeta }$.

arXiv comment15 pages. Results generalized to include both real and complex K-theory. To appear in J. K-Theory