Suppose that B is a G-Banach algebra over 𝔽 = ℝ or ℂ, X is a finite dimensional compact metric space, ζ : P → X is a standard principal G-bundle, and Aζ = Γ(X,P xG B) is the associated algebra of sections. We produce a spectral sequence which converges to π∗(GLoAζ) with
E_2p,q ≅ Ȟp(X ; πq(GLoB)).
A related spectral sequence converging to K∗+1(Aζ) (the real or complex topological K-theory) allows us to conclude that if B is Bott-stable, (i.e., if π∗(GLoB) → K∗+1(B) is an isomorphism for all ∗ > 0) then so is Aζ.
Algebra | Analysis | Physical Sciences and Mathematics
E. Dror Farjouin and C. Schochet, Spaces of sections of Banach algebra bundles, Journal of K-Theory 10.2 (2012), 279-298. doi:10.1017/is012002001jkt183