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# Strict Comparison of Positive Elements in Multiplier Algebras

Published:2016-06-28
Printed: Apr 2017
• Victor Kaftal,
Department of Mathematics , University of Cincinnati, P. O. Box 210025 , Cincinnati, OH, 45221-0025 , USA
• Ping Wong Ng,
Department of Mathematics , University of Louisiana, 217 Maxim D. Doucet Hall , P.O. Box 43568, Lafayette, Louisiana, 70504-3568 , USA
• Shuang Zhang,
Department of Mathematics , University of Cincinnati, P.O. Box 210025 , Cincinnati, OH, 45221-0025, USA
 Format: LaTeX MathJax PDF

## Abstract

Main result: If a C*-algebra $\mathcal{A}$ is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra $\operatorname{\mathcal{M}}(\mathcal{A})$ also has strict comparison of positive elements by traces. The same results holds if finitely many extremal traces" is replaced by quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If $\mathcal{A}$ is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.
 Keywords: strict comparison, bi-diagonal form, positive combinations
 MSC Classifications: 46L05 - General theory of $C^*$-algebras 46L35 - Classifications of $C^*$-algebras 46L45 - Decomposition theory for $C^*$-algebras 47C15 - Operators in $C^*$- or von Neumann algebras

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