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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 $C^{*}$-algebra

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

Keywords:  $C^*$-algebras, multiplier algebras, unitary group

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