Abstract view
Strict Comparison of Positive Elements in Multiplier Algebras


Published:20160628
Printed: Apr 2017
Victor Kaftal,
Department of Mathematics , University of Cincinnati, P. O. Box 210025 , Cincinnati, OH, 452210025 , USA
Ping Wong Ng,
Department of Mathematics , University of Louisiana, 217 Maxim D. Doucet Hall , P.O. Box 43568, Lafayette, Louisiana, 705043568 , USA
Shuang Zhang,
Department of Mathematics , University of Cincinnati, P.O. Box 210025 , Cincinnati, OH, 452210025, USA
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 bidiagonal 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.