Journal of Operator Theory
Volume 73, Issue 1, Winter 2015 pp. 187-210.
Strict comparison of projections and positive combinations of projections in certain multiplier algebras
Authors:
Victor Kaftal (1), Ping W. Ng, (2) and Shuang Zhang (3)
Author institution:(1) Department of Mathematical Sciences,
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 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: 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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