Spaces of sections of Banach algebra bundles

Suppose that B is a G-Banach algebra over F = R or C, X is a finite dimensional compact metric space, zeta : P -> X is a standard principal G-bundle, and A(zeta) = Gamma(X, P x(G) B) is the associated algebra of sections. We produce a spectral sequence which converges to pi(*)(GL(o)A(zeta)) with E--p,q(2) congruent to 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) -> K*+1(B) is an isomorphism for all * > 0) then so is A(zeta).