Journal of Operator Theory

Volume 79, Issue 2, Spring 2018 pp. 419-462.

The minimal ideal in multiplier algebras

Authors: Victor Kaftal (1) P.W. Ng (2) and Shuang Zhang (3)
Author institution: (1) Department of Mathematics, University of Cincinnati, P. O. Box 210025, Cincinnati, OH, 45221-0025, U.S.A.
(2) Department of Mathematics, University of Louisiana, 217 Maxim D. Doucet Hall, P.O. Box 43568, Lafayette, Louisiana, 70504-3568, U.S.A.
(3) Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, OH, 45221-0025, U.S.A.

Summary: When $\mathcal{A}$ is a simple, $\sigma$ -unital, non-unital, non-elementary $C^*$ -algebra, let $I_{\mathrm{min}}$ denote the intersection of the ideals of $\mathcal M(\mathcal A)$ that properly contain $\mathcal{A}$ . $I_{\mathrm{min}}$ coincides with the ideal defined by Lin. We prove that $I_{\mathrm{min}}\ne\mathcal{A}$ for several categories of $C^*$ -algebras. If $I_{\mathrm{min}}\ne\mathcal{A}$ , then $I_{\mathrm{min}}/\mathcal{A}$ is purely infinite and simple. If $\mathcal{A}$ has strict comparison of positive elements by traces then $I_{\mathrm{min}}=I_{\mathrm{cont}}$ , the closure of the linear span of the elements $A\in \mathcal M(\mathcal A)_+$ such that the evaluation map $\widehat A(\tau)=\tau(A)$ is continuous. In particular, $I_{\mathrm{min}}\ne I_{\mathrm{cont}}$ for certain Villadsen's AH-algebras.

DOI: http://dx.doi.org/10.7900/jot.2017may12.2161
Keywords: multiplier algebras, minimal ideals, strict comparison, Villadsen AH-algebras

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