Spaces of sections of Banach algebra bundles

TL;DR

This paper develops spectral sequences to analyze the homotopy groups and K-theory of sections of Banach algebra bundles over finite-dimensional compact spaces, linking properties of the fiber algebra to the bundle algebra.

Contribution

It introduces spectral sequences that connect the homotopy and K-theory of bundle sections with the base space and fiber algebra, extending Bott-stability results.

Findings

Spectral sequence converging to _*(GL_o A_eta) with E^2-term involving Čech cohomology.

Spectral sequence converging to K_{*+1}(A_eta) for topological K-theory.

If the fiber algebra B is Bott-stable, then the algebra of sections A_eta is also Bott-stable.

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_*(GL_o A_\zeta) $ with [E^2_{-p,q} \cong \check{H}^p(X ; \pi_q(GL_o B)).] A related spectral sequence converging to $\K_{*+1}(A_\zeta)$ (the real or complex topological $K$-theory) allows us to conclude that if $B$ is Bott-stable, (i.e., if $ \pi_*(GL_o B) \to \K_{*+1}(B)$ is an isomorphism for all $*>0$) then so is $A_\zeta$.