  Suppose that $B$ is a $G$-Banach algebra over $\mathbb{F} = \mathbb{R}$ or
$\mathbb{C}$, $X$ is a finite dimensional compact metric space, $\zeta : P \to
X$ is a standard principal $G$-bundle, and $A_\zeta = \Gamma (X, P \times_G B)$
is the associated algebra of sections.
  We produce a spectral sequence which converges to $\pi_*(GL_o A_\zeta) $ with
[E^2_{-p,q} \cong \check{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) \to
\K_{*+1}(B)$ is an isomorphism for all $*>0$) then so is $A_\zeta$.
