Fundamental Group of Simple C*-algebras with Unique Trace III

Nawata, Norio

Canadian journal of mathematics, Tome 64 (2012) no. 3, pp. 573-587

Résumé

We introduce the fundamental group $\mathcal{F}\left( A \right)$ of a simple $\sigma $ -unital ${{C}^{*}}$ –algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple ${{C}^{*}}$ -algebras with Unique Trace I and II by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless ${{C}^{*}}$ -algebras. We show that there exist separable stably projectionless ${{C}^{*}}$ -algebras such that their fundamental groups are equal to $\mathbb{R}_{+}^{\times }$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Détail

DOI : 10.4153/CJM-2011-052-0

Mots-clés : 46L05, 46L08, 46L35, fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function