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Perforation conditions and almost algebraic order in Cuntz semigroups

Part of: Lattices Selfadjoint operator algebras $K$-theory and operator algebras Special categories Basic linear algebra

Published online by Cambridge University Press:  18 April 2018

Ramon Antoine ,
Francesc Perera  and
Henning Petzka
Ramon Antoine
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain (ramon@mat.uab.cat; perera@mat.uab.cat)
Francesc Perera
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain (ramon@mat.uab.cat; perera@mat.uab.cat)
Henning Petzka
Affiliation:
Mathematisches Institut, Universität Münster, Einsteinstraße 62, 48149 Münster, Germany (petzka@uni-muenster.de)
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Abstract

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.

Keywords

Cuntz semigrouptensor productC*-algebra

MSC classification

Secondary: 06B35: Continuous lattices and posets, applications 15A69: Multilinear algebra, tensor products 46L05: General theory of $C^*$-algebras 46L35: Classifications of $C^*$-algebras 46L80: $K$-theory and operator algebras (including cyclic theory) 18B35: Preorders, orders and lattices (viewed as categories) 19K14: $K_0$ as an ordered group, traces 46L06: Tensor products of $C^*$-algebras
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 4 , August 2018 , pp. 669 - 702
Copyright
Copyright © Royal Society of Edinburgh 2018 

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