SimpleC*-algebra generated by isometries

Cuntz, Joachim

doi:10.1007/bf01625776

Cited by 1,014 publications

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“…Recall that the Cuntz algebra O ∞ is the universal unital C * -algebra generated by a set of isometries {s i | i ∈ N} with mutually orthogonal range projections s i s * i [15]. Observe that O ∞ is a unital C * -algebra, so that O ∞ -stabilization preserves unitality (unlike K-stabilization).…”

Section: Our Resultsmentioning

confidence: 99%

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: Our Resultsmentioning

confidence: 99%

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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“…Asking perfect reconstruction then amounts to a decomposition of the identity operator. These exact requirements are met by representations of relations known in Operator Algebra theory as the Cuntz relations; see [Cun77,? ] and Figure 1.…”

Section: Benefits Of Entropy Encodingmentioning

confidence: 99%

Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

Jørgensen

Song

2009

Acta Appl Math

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The purpose of this paper is to show that algorithms in a diverse set of applications may be cast in the context of relations on a finite set of operators in Hilbert space. The Cuntz relations for a finite set of isometries form a prototype of these relations. Such applications as entropy encoding, analysis of correlation matrices (Karhunen-Loève), fractional Brownian motion, and fractals more generally, admit multi-scales. In signal/image processing, this may be implemented with recursive algorithms using subdivisions of frequency-bands; and in fractals with scale similarity. In Karhunen-Loève analysis, we introduce a diagionalization procedure; and we show how the Hilbert space formulation offers a unifying approach; as well as suggesting new results.

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“…For N ≥ 2, let O N be the Cuntz algebra( [4]), that is, it is a C * -algebra which is universally generated by generators s 1 , . .…”

Section: Permutative Representationsmentioning

confidence: 99%

Polynomial Endomorphisms of the Cuntz Algebras Arising from Permutations: I ? General Theory

Kawamura

2005

Lett Math Phys

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SimpleC*-algebra generated by isometries

Cited by 1,014 publications

References 10 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

Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

Polynomial Endomorphisms of the Cuntz Algebras Arising from Permutations: I ? General Theory

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