Classification of Nuclear, Simple C*-algebras

Rørdam, Mikael

doi:10.1007/978-3-662-04825-2_1

Cited by 173 publications

(207 citation statements)

References 90 publications

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“…Lemma 1.4). The range of applicability of this trick is much broader than the case explored here (see, for instance, Proposition 1.1.2 of [53]). The author is grateful to D. Enders for pointing out that Proposition 2.2, that uses this trick, can be generalized to all C * -algebras of the form A⊗B with B properly infinite.…”

Section: Introductionmentioning

confidence: 86%

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Mahanta

2015

Math Phys Anal Geom

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Abstract. We develop an algebraic formalism for topological T-duality. More precisely, we show that topological T-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C * -algebra D. Then we show that there is a functorial topological K-theory symmetric spectrum construction K top Σ (−) on the category of separable C * -algebras, such that K top Σ (D) is an algebra over this operad; moreover, K top Σ (A⊗D) is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C * -algebras. We also show that O ∞ -stable C * -algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of ax + b-semigroup C * -algebras coming from number theory and that of O ∞ -stabilized noncommutative tori. IntroductionWithin the category of separable C * -algebras SC * the stable ones, i.e., those A ∈ SC * satisfying A⊗K ∼ = A, play a privileged role. For instance, it is known that they satisfy the Karoubi conjecture and appear very naturally in the context of twisted K-theory. The results in this article demonstrate that O ∞ -stable separable C * -algebras, i.e., those A ∈ SC * satisfying A⊗O ∞ ∼ = A, deserve a similar prominent status. Moreover, the Cuntz algebra O ∞ is strongly self-absorbing, which has several interesting ramifications.Ever since its inception by Kontsevich [33] the homological mirror symmetry conjecture has promoted rich interaction between geometry, algebra, and (higher) category theory. Mirror symmetry is related to T-duality via the Strominger-Yau-Zaslow conjecture [61] and hence we believe that it is worthwhile to have an algebraic formalism for T-duality at our disposal. One aspect of this theory is the Bunke-Schick topological T-duality, which has received a lot of attention in the mathematical literature. It is insensitive to subtle geometric structures but its mathematical underpinnings are very well understood [8,7,9]. One of the objectives of this article is to develop an algebraic formalism for topological T-duality relating it to the theory of noncommutative motives [34,35,65,45]. Along the way we obtain several interesting applications to algebraic K-theory and K-regularity of C * -algebras. The novelty 2010 Mathematics Subject Classification. 46L85, 19Dxx, 46L80, 57R56.

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Section: Introductionmentioning

confidence: 86%

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Mahanta

2015

Math Phys Anal Geom

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“…We refer to [14], [11], and [41] for details. Nuclearity is the analogue of amenability for C * -algebras.…”

Section: Other Important Propertiesmentioning

confidence: 99%

“…A simple separable nuclear purely infinite C * -algebra is called a Kirchberg algebra. The theorem states that Kirchberg algebras satisfying the UCT are classifed up to stable isomorphism by their K-theory (see [41]). …”

Section: Other Important Propertiesmentioning

confidence: 99%

C⁎-algebras associated to graphs of groups

Brownlowe

Mundey

Pask

et al. 2017

Advances in Mathematics

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Abstract. To a large class of graphs of groups we associate a C * -algebra universal for generators and relations. We show that this C * -algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C * -algebras associated to generalised BaumslagSolitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.

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“…Direct limits of full matrix algebras are called UHF algebras and they were classified by Glimm (the unital case) and Dixmier (the general case) in the 1960s. This was the start of the Elliott classification program of separable unital C * -algebras (see [48], [19]). …”

Section: Uhf (Uniformly Hyperfinite) Algebras For Each Natural Numbermentioning

confidence: 99%

“…This class of C*-algebras is much more extensive than the class of UHF algebras. Elliott's classification of unital separable AF algebras by K-theoretic invariant K 0 (see [48]) marked the beginning of Elliott program for classification of C*-algebras.…”

Section: Uhf (Uniformly Hyperfinite) Algebras For Each Natural Numbermentioning

confidence: 99%

Set theory and operator algebras

Farah

Wofsey

Appalachian Set Theory 2006–2012

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Classification of Nuclear, Simple C*-algebras

Cited by 173 publications

References 90 publications

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

C⁎-algebras associated to graphs of groups

Set theory and operator algebras

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