Journal of Operator Theory
Volume 68, Issue 2, Fall 2012 pp. 549-565.
On the structure of the projective unitary group of the multiplier algebra of a simple stable nuclear C∗-algebraAuthors: P.W. Ng (1) and Efren Ruiz (2)
Author institution: (1) Department of Mathematics, University of Louisiana at Lafayette 217 Maxim D. Doucet Hall, P. O. Box 41010, Lafayette, Louisiana, 70504--1010 U.S.A.
(2) Department of Mathematics, University of Hawai'i, Hilo, 200 W. Kawili St. Hilo, Hawai'i, 96720 U.S.A.
Summary: Let A be a simple unital separable C∗-algebra. Let U(\Mul(A⊗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(\Mul(A⊗K))/T, given the quotient topology induced by the strict topology on U(\Mul(A⊗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 \Mul(A⊗K) has the AF-property in the strict topology; i.e., there exists a unital AF-C∗-subalgebra C⊂\Mul(A⊗K) such that C is strictly dense in \Mul(A⊗K). We also observe that there are Kaplansky density-type theorems for \Mul(A⊗K) with the strict topology. This, together with the preceding result, imply that if A is nuclear then U(\Mul(A⊗K))/T is the topological closure of an increasing union of simple compact topological subgroups.
Contents Full-Text PDF