Journal of Operator Theory

Volume 73, Issue 1, Summer 2015 pp. 187-210.

Strict comparison of projections and positive combinations of projections in certain multiplier algebras

Authors: (1) Victor Kaftal, (2) Ping W. Ng, (3) Shuang Zhang
Author institution: (1) Department of Mathematical Sciences, University of Cincinnati, P. O. Box 210025, Cincinnati, OH, 45221-0025, U.S.A.
(2) University of Louisiana, 217 Maxim D. Doucet Hall, P.O. Box 41010, Lafayette, Louisiana, 70504-1010, U.S.A.
(3) Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH, 45221-0025, U.S.A.

Summary: $\DeclareMathOperator{\M}{\mathcal M(\mathcal A \otimes \mathcal K)}$ In this paper we investigate whether positive elements in the multiplier algebras of certain finite $C^*$-algebras can be written as finite linear combinations of projections with positive coefficients (PCP). Our focus is on the category of underlying $C^*$-algebras that are separable, simple, with real rank zero, stable rank one, finitely many extreme traces, and strict comparison of projections by the traces. We prove that the strict comparison of projections holds also in the multiplier algebra $\M$. Based on this result and under the additional hypothesis that $\M$ has real rank zero, we characterize which positive elements of $\M$ are of PCP.

DOI: http://dx.doi.org/10.7900/jot.2013nov05.2014
Keywords: Finite $C^*$-algebras, multiplier algebras, strict comparison of projections, positive combinations of projections.

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