CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

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 strict comparison, bi-diagonal form, positive combinations
MSC Classifications: 46L05, 46L35, 46L45, 47C15 show english descriptions General theory of $C^*$-algebras
Classifications of $C^*$-algebras
Decomposition theory for $C^*$-algebras
Operators in $C^*$- or von Neumann algebras
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
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/