On the structure of the projective unitary group of the multiplier algebra of a simple stable nuclear C*-algebra

摘要

Let A be a simple unital separable C*-algebra. Let U(M(A{position indicator}K)) be the unitary group of the multiplier algebra of the stabilization of A, with the strict topology; and let T be the subgroup of scalar unitaries. We prove that U(M(A{position indicator}K))/T, given the quotient topology induced by the strict topology on U(M(A position indicator}K)), is a simple topological group. We also give a characterization of nuclearity of A. In particular, we show that A is nuclear if and only if M(A{position indicator}K) has the AF-property in the strict topology; i.e., there exists a unital AF-C*-subalgebra C ? M(A{position indicator}K) such that C is strictly dense inM(A{position indicator}K). We also observe that there are Kaplansky density-type theorems for M (A{position indicator}K) with the strict topology. This, together with the preceding result, imply that if A is nuclear then U(M(A{position indicator}K))/T is the topological closure of an increasing union of simple compact topological subgroups.