Spaces of sections of Banach algebra bundles
Emmanuel Dror Farjoun, Claude L. Schochet

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 is a -Banach algebra over or , is a finite dimensional compact metric space, is a standard principal -bundle, and is the associated algebra of sections. We produce a spectral sequence which converges to with [E^2_{-p,q} \cong \check{H}^p(X ; \pi_q(GL_o B)).] A related spectral sequence converging to (the real or complex topological -theory) allows us to conclude that if is Bott-stable, (i.e., if is an isomorphism for all ) then so is .
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