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 A is a simple, σ-unital, non-unital, non-elementary C∗-algebra,
let Imin denote the intersection of the ideals of M(A) that properly contain A. Imin coincides with the ideal defined by Lin. We prove that Imin≠A for several categories of C∗-algebras. If Imin≠A, then Imin/A is purely infinite and simple. If A has strict comparison of positive elements by traces then Imin=Icont, the closure of the linear span of the elements A∈M(A)+ such that the evaluation map ˆA(τ)=τ(A) is continuous. In particular, Imin≠Icont 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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