On the homotopy type of the unitary group and the Grassmann space of purely infinite simple \(C^*\)-algebras (Q1598948)

The paper provides some topological information about purely infinite simple \(C^*\)-algebras in terms of \(K\)-theory. Let \(A\) be one such algebra and \(\widetilde A\) be its unitalization. It is shown that the homotopy groups of the unitary group and the space of projections of \(\widetilde A\) are isomorphic to either \(K_0(A)\) or \(K_1(A)\). It follows from this that \(K_0(A)= K_1(A)=0\) if and only if the unitary group of \(\widetilde A\) is a contractible topological space. The same results are also true if \(\widetilde A\) is replaced by the associated generalized Calkin algebra.