Spaces of sections of Banach algebra bundles

Suppose that $B$ is a $G$-Banach algebra over $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, $X$ is a finite dimensional compact metric space, $ζ: P \to X$ is a standard principal $G$-bundle, and $A_ζ= Γ(X, P \times_G B)$ is the associated algebra of sections. We produce a spectral sequence which converges to $π_*(GL_o A_ζ)$ with [E^2_{-p,q} \cong \check{H}^p(X ; π_q(GL_o B)).] A related spectral sequence converging to $\K_{*+1}(A_ζ)$ (the real or complex topological $K$-theory) allows us to conclude that if $B$ is Bott-stable, (i.e., if $ π_*(GL_o B) \to \K_{*+1}(B)$ is an isomorphism for all $*>0$) then so is $A_ζ$.

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