Regularity for stably projectionless, simple C*-algebras

Aaron Peter Tikuisis

doi:10.1016/j.jfa.2012.05.020

Regularity for stably projectionless, simple C^*-algebras

Aaron Peter Tikuisis

Computing Science

Deutsche Forschungsgesellschaft DFG

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

10 Downloads (Pure)

Abstract

This paper explores the following regularity properties and their relationships for simple, not-necessarily-unital C*-algebras:
(i) Jiang-Su stability,
(ii) unperforation in the Cuntz semigroup, and
(iii) slow dimension growth (applying only in the case that the C*-algebra is approximately subhomogeneous).
An example is given of a simple, separable, nuclear, stably projectionless C*-algebra whose Cuntz semigroup is not almost unperforated. This example is in fact approximately subhomogeneous. It is also shown that, in contrast to this example, when an approximately subhomogeneous simple C*-algebra has slow dimension growth, its Cuntz semigroup is necessarily almost unperforated.

Original language	English
Pages (from-to)	1382–1407
Number of pages	26
Journal	Journal of Functional Analysis
Volume	263
Issue number	5
Early online date	18 Jun 2012
DOIs	https://doi.org/10.1016/j.jfa.2012.05.020
Publication status	Published - 1 Sept 2012

Keywords

Stably projectionless C*-algebras
Cuntz semigroup
Jiang-Su algebra
approximately subhomogeneous C*-algebras
slow dimension growth

Access to Document

10.1016/j.jfa.2012.05.020

Projless regularity
NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Functional Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in JOURNAL OF FUNCTIONAL ANALYSIS, [VOL 263, ISSUE 5, (2012)] DOI: 10.1016/j.jfa.2012.05.020
Accepted author manuscript, 376 KB

Cite this

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title = "Regularity for stably projectionless, simple C*-algebras",

abstract = "This paper explores the following regularity properties and their relationships for simple, not-necessarily-unital C*-algebras: (i) Jiang-Su stability, (ii) unperforation in the Cuntz semigroup, and (iii) slow dimension growth (applying only in the case that the C*-algebra is approximately subhomogeneous). An example is given of a simple, separable, nuclear, stably projectionless C*-algebra whose Cuntz semigroup is not almost unperforated. This example is in fact approximately subhomogeneous. It is also shown that, in contrast to this example, when an approximately subhomogeneous simple C*-algebra has slow dimension growth, its Cuntz semigroup is necessarily almost unperforated.",

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N2 - This paper explores the following regularity properties and their relationships for simple, not-necessarily-unital C*-algebras: (i) Jiang-Su stability, (ii) unperforation in the Cuntz semigroup, and (iii) slow dimension growth (applying only in the case that the C*-algebra is approximately subhomogeneous). An example is given of a simple, separable, nuclear, stably projectionless C*-algebra whose Cuntz semigroup is not almost unperforated. This example is in fact approximately subhomogeneous. It is also shown that, in contrast to this example, when an approximately subhomogeneous simple C*-algebra has slow dimension growth, its Cuntz semigroup is necessarily almost unperforated.

AB - This paper explores the following regularity properties and their relationships for simple, not-necessarily-unital C*-algebras: (i) Jiang-Su stability, (ii) unperforation in the Cuntz semigroup, and (iii) slow dimension growth (applying only in the case that the C*-algebra is approximately subhomogeneous). An example is given of a simple, separable, nuclear, stably projectionless C*-algebra whose Cuntz semigroup is not almost unperforated. This example is in fact approximately subhomogeneous. It is also shown that, in contrast to this example, when an approximately subhomogeneous simple C*-algebra has slow dimension growth, its Cuntz semigroup is necessarily almost unperforated.

KW - Stably projectionless C-algebras

KW - Cuntz semigroup

KW - Jiang-Su algebra

KW - approximately subhomogeneous C-algebras

KW - slow dimension growth

U2 - 10.1016/j.jfa.2012.05.020

DO - 10.1016/j.jfa.2012.05.020

M3 - Article

SN - 0022-1236

VL - 263

SP - 1382

EP - 1407

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 5

ER -