Topology on s−1s for banach algebras

In this note we show that Quillenʹs S−1 S-construction on the category of finitely generated projective modules over a Banach algebra A with identity has a topological enrichment S−1Stop(Λ) and B(S−1Stop(Λ)) has the homotopy type of KOΛ × BGLtopΛ. By applying the singular complex functor, we obtain a first quadrant spectral E1p,q(Λ) = KalgqΛ(Δp), where Λ(Δp) is the ring of continuous Λ-valued functions on Δp, that converges to Ktopp + qΛ = πp + q(K0Λ × BGLtopΛ) for p + q ⩾ 0.