Topology on \(S^{-1}S\) for Banach algebras (Q1922753)

Let \(\mathcal P\) be the category of finitely generated projective modules over a Banach algebra with identity \(\Lambda\), and \(\Lambda(\Delta^p)\) the ring of continuous \(\Lambda\)-valued functions on \(\Delta^p\). Quillen's \(S^{-1}S\)-construction provides a model for the algebraic \(K\)-theory of \(\Lambda\) [\textit{D. Grayson}, Lect. Notes Math. 551, 217-240 (1976; Zbl 0362.18015)]. This note shows that there is a topological enrichment \(S^{-1}S^{\text{top}}({\mathcal P})\) of the category \(S^{-1}S({\mathcal P})\) which computes the topological \(K\)-theory of \(\Lambda\): \[ K^{\text{top}}_p(\Lambda)\cong \pi_p S^{-1} S^{\text{top}}({\mathcal P}). \] The obvious functor \(S^{-1}S({\mathcal P})\to S^{-1} S^{\text{top}}({\mathcal P})\) induces the expected map from the algebraic to the topological \(K\)-groups of \(\Lambda\). Besides, this construction gives rise to a spectral sequence \(E^1_{p,q}(\Lambda)=K^{\text{alg}}_q \Lambda(\Delta^p)\) that converges to \(K^{\text{top}}_{p+q}(\Lambda)\).