Trivial \(K_ 1\)-flow of AF algebras and finite von Neumann algebras
A \(C^*\)-algebra \({\mathcal A}\) is said to have a trivial \(K_ 1\)-flow if \(K_ 1({\mathcal B})=0\) for any hereditary \(C^*\)-subalgebra \({\mathcal B}\) of M(\({\mathcal A})\). We prove that if \({\mathcal A}\) is a \(\sigma\)-unital AF algebra, then \({\mathcal A}\) has a trivial \(K_ 1\)-flow, and that if \({\mathcal A}\) is a finite von Neumann algebra, then \({\mathcal A}\otimes {\mathcal K}\) has a trivial \(K_ 1\)-flow. If the multiplier algebra of \({\mathcal A}\) has the FS property, then the unitary group of \(\tilde {\mathcal B}\) is connected for any hereditary \(C^*\)-subalgebra \({\mathcal B}\) of M(\({\mathcal A})\). We also prove that if \({\mathcal A}\) is either a \(\sigma\)-unital nonunital purely infinite simple \(C^*\)-algebra or a nonunital \(C^*\)-algebra stably isomorphic to a Bunce-Deddens algebra, then \(K_ 1({\mathcal B})=0\) for any hereditary \(C^*\)-subalgebra \({\mathcal B}\) of M(\({\mathcal A})\) not contained in \({\mathcal A}\).
scientific article; zbMATH DE number 4101890
scientific article