Spaces of sections of Banach algebra bundles

Suppose that B is a G-Banach algebra over F = R or CX is a finite dimensional compact metric space, ξ : P → X is a standard principal G-bundle, and A ξ = Λ(X,P × G B) is the associated algebra of sections. We produce a spectral sequence which converges to π _*(GLoAξ) with E ² _-p,q ≃ Ĥ ^p(X;π _q(GL _oB)) A related spectral sequence converging to K _*+1(A _ξ) (the real or complex topological K-theory) allows us to conclude that if B is Bott-stable, (i.e., if π _*(GL _o B) → K _*+1(B) is an isomorphism for all * > 0) then so is A _ξ.