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# On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras

Published:2015-03-18
Printed: Jun 2015
• Aaron Peter Tikuisis,
Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom
• Andrew Toms,
Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, IN 47907, USA
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## Abstract

We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple $\mathrm{C}^*$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first-named author, this shows that slow dimension growth coincides with $\mathcal Z$-stability, for approximately subhomogeneous algebras.
 Keywords: nuclear C*-algebras, Cuntz semigroup, dimension functions, stably projectionless C*-algebras, approximately subhomogeneous C*-algebras, slow dimension growth
 MSC Classifications: 46L35 - Classifications of $C^*$-algebras 46L05 - General theory of $C^*$-algebras 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 47L40 - Limit algebras, subalgebras of $C^*$-algebras 46L85 - Noncommutative topology [See also 58B32, 58B34, 58J22]

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