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

Published:2011-08-03
Printed: Jun 2012
• Norio Nawata,
Graduate School of Mathematics, Kyushu University, Motooka, Fukuoka, 819-0395, Japan
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## Abstract

We introduce the fundamental group ${\mathcal F}(A)$ 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.
 Keywords: fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function
 MSC Classifications: 46L05 - General theory of $C^*$-algebras 46L08 - $C^*$-modules 46L35 - Classifications of $C^*$-algebras