Strict Comparison of Positive Elements in Multiplier Algebras

Kaftal, Victor ; Ng, Ping Wong ; Zhang, Shuang

Canadian journal of mathematics, Tome 69 (2017) no. 2, pp. 373-407

Résumé

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 $\mathcal{M}\left( \mathcal{A} \right)$ 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. As 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.

Détail

DOI : 10.4153/CJM-2016-015-3

Mots-clés : 46L05, 46L35, 46L45, 47C15, strict comparison, bi-diagonal form, positive combinations