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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 IminA for several categories of C-algebras. If IminA, 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 AM(A)+ such that the evaluation map ˆA(τ)=τ(A) is continuous. In particular, IminIcont 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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