Projective Modules over Higher-Dimensional Non-Commutative Tori

Rieffel, Marc A.

doi:10.4153/cjm-1988-012-9

Cited by 226 publications

(168 citation statements)

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“…In [19] Feichtinger and Luef give a detailed answer to when (5.10) holds in the setting of R n , see also [17,21] for related results. The FIGA was first proved by Rieffel [38] for generators g, h in the Schwartz-Bruhat space S(G). Rieffel's proof uses the Poisson summation formula and also holds for the non-separable case with closed subgroups in G × G; it is also possbile to give an argument based on Janssen's proof for (lattice) Gabor systems in L 2 (R) [30,31].…”

Section: The Janssen Representations Of the Frame Operatormentioning

confidence: 99%

“…Rieffel [38] proved in 1988 a weak form of the Janssen representation called the fundamental identity in Gabor analysis (FIGA) for Gabor systems in L 2 (G) with modulations and translations along a closed subgroup in G× G, where G is a second countable LCA group and G its dual group. Most other structure and duality results assume Gabor systems in L 2 (G) with modulations and translations along discrete and co-compact subgroups (also called uniform lattices), e.g., the Wexler-Raz biorthogonal relations for such uniform lattice Gabor systems appear implicitly in the work of Gröchenig [23].…”

Section: Introductionmentioning

confidence: 99%

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Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.

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Section: The Janssen Representations Of the Frame Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Co-compact Gabor Systems on Locally Compact Abelian Groups

Jakobsen

Lemvig

2015

J Fourier Anal Appl

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“…It follows from [51] that for irrational θ the projections in A n θ generate all of K 0 (A n θ⊗ O ∞ ) and…”

Section: 2mentioning

confidence: 99%

“…We just determined the isomorphism type of the algebraic K-theory groups of A n θ⊗ O ∞ . One can also describe the elements in these groups using Rieffel's results in [51].…”

Section: 2mentioning

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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“…the same properties for θ = 0). So equipped C ∞ (M θ , S) is in particular an involutive bimodule with a right-hermitian structure [46], [29].…”

mentioning

confidence: 99%

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Connes¹,

Dubois-Violette²

2002

Commun Math Phys

167

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We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations S 3 u of the standard 3-sphere S 3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space R 4 . For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u only depend on two parameters and are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. It follows that different S 3 u can span the same R 4 u . This equivalence generates a foliation of the parameter space Σ. This foliation admits singular leaves reduced to a point. These critical points are either isolated or fall in two 1-parameter families C± ⊂ Σ. Up to the simple operation of taking the fixed algebra by an involution, these two families are identical and we concentrate here on C+. For u ∈ C+ the above isomorphism with the Sklyanin algebra breaks down and the corresponding algebras are special cases of θ-deformations, a notion which we generalize in any dimension and various contexts, and study in some details. Here, and this point is crucial, the dimension is not an artifact, i.e. the dimension of the classical model, but is the Hochschild dimension of the corresponding algebra which remains constant during the deformation. Besides the standard noncommutative tori, examples of θ-deformations include the recently defined noncommutative 4-sphere S 4 θ as well as m-dimensional generalizations, noncommutative versions of spaces R m and quantum groups which are deformations of various classical groups. We develop general tools such as the twisting of the Clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative 2n-dimensional spherical manifolds S 2n θ . A key result is the differential self-duality properties of these projections which generalize the self-duality of the round instanton.

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Projective Modules over Higher-Dimensional Non-Commutative Tori

Cited by 226 publications

References 37 publications

Co-compact Gabor Systems on Locally Compact Abelian Groups

Co-compact Gabor Systems on Locally Compact Abelian Groups

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

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