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
Contents
Full-Text PDF