We study groupoid actions on left cancellative small categories and their associated Zappa-Sz'ep products. We show that certain left cancellative small categories with nice length functions can be seen as Zappa-Sz'ep products. We compute the associated tight groupoids, characterizing important properties of them, like being Hausdorff, effective and minimal. Finally, we determine amenability of the tight groupoid under mild, reasonable hypotheses.
{"title":"Zappa–Szép products for partial actions of groupoids on left cancellative small categories","authors":"E. Ortega, E. Pardo","doi":"10.4171/jncg/518","DOIUrl":"https://doi.org/10.4171/jncg/518","url":null,"abstract":"We study groupoid actions on left cancellative small categories and their associated Zappa-Sz'ep products. We show that certain left cancellative small categories with nice length functions can be seen as Zappa-Sz'ep products. We compute the associated tight groupoids, characterizing important properties of them, like being Hausdorff, effective and minimal. Finally, we determine amenability of the tight groupoid under mild, reasonable hypotheses.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44606738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On filtered manifolds one can define a different notion of order for the differential operators. In this paper, we use generalized fixed point algebras to construct a pseudodifferential extension that reflects this behaviour. In the corresponding calculus, the principal symbol of an operator is a family of operators acting on certain nilpotent Lie groups. The role of ellipticity as a Fredholm condition is replaced by the Rockland condition on these groups. Our approach allows to understand this in terms of the representation of the corresponding algebra of principal symbols. Moreover, we compute the $K$-theory of this algebra.
{"title":"Pseudodifferential operators on filtered manifolds as generalized fixed points","authors":"E. Ewert","doi":"10.4171/jncg/502","DOIUrl":"https://doi.org/10.4171/jncg/502","url":null,"abstract":"On filtered manifolds one can define a different notion of order for the differential operators. In this paper, we use generalized fixed point algebras to construct a pseudodifferential extension that reflects this behaviour. In the corresponding calculus, the principal symbol of an operator is a family of operators acting on certain nilpotent Lie groups. The role of ellipticity as a Fredholm condition is replaced by the Rockland condition on these groups. Our approach allows to understand this in terms of the representation of the corresponding algebra of principal symbols. Moreover, we compute the $K$-theory of this algebra.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47159976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the Open Mapping Theorem for octonionic slice regular functions. Moreover, it opens the path for a possible use of slice regular functions in the study of almost-complex structures in eight dimensions. Acknowledgements. This work was partly supported by GNSAGA of INdAM, by the INdAM project “Hypercomplex function theory and applications” and by the PRIN 2017 project “Real and Complex Manifolds” of the Italian Ministry of Education (MIUR). The third author is also supported by Finanzi-amento Premiale FOE 2014 “Splines for accUrate NumeRics: adaptIve models for Simulation Environ-ments” of MIUR. The authors are grateful to the anonymous referee for the precious suggestions.
{"title":"Slice regular functions and orthogonal complex structures over $mathbb{R}^8$","authors":"R. Ghiloni, A. Perotti, C. Stoppato","doi":"10.4171/jncg/452","DOIUrl":"https://doi.org/10.4171/jncg/452","url":null,"abstract":"This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the Open Mapping Theorem for octonionic slice regular functions. Moreover, it opens the path for a possible use of slice regular functions in the study of almost-complex structures in eight dimensions. Acknowledgements. This work was partly supported by GNSAGA of INdAM, by the INdAM project “Hypercomplex function theory and applications” and by the PRIN 2017 project “Real and Complex Manifolds” of the Italian Ministry of Education (MIUR). The third author is also supported by Finanzi-amento Premiale FOE 2014 “Splines for accUrate NumeRics: adaptIve models for Simulation Environ-ments” of MIUR. The authors are grateful to the anonymous referee for the precious suggestions.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48763948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a slice--regular quaternionic function $f,$ the classical exponential function $exp f$ is not slice--regular in general. An alternative definition of exponential function, the $*$-exponential $exp_*$, was given: if $f$ is a slice--regular function, then $exp_*(f)$ is a slice--regular function as well. The study of a $*$-logarithm $log_*(f)$ of a slice--regular function $f$ becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a $log_*(f)$ depends only on the structure of the zero set of the vectorial part $f_v$ of the slice--regular function $f=f_0+f_v$, besides the topology of its domain of definition. We also show that, locally, every slice--regular nonvanishing function has a $*$-logarithm and, at the end, we present an example of a nonvanishing slice--regular function on a ball which does not admit a $*$-logarithm on that ball.
{"title":"On a definition of logarithm of quaternionic functions","authors":"G. Gentili, Jasna Prezelj, Fabio Vlacci","doi":"10.4171/jncg/514","DOIUrl":"https://doi.org/10.4171/jncg/514","url":null,"abstract":"For a slice--regular quaternionic function $f,$ the classical exponential function $exp f$ is not slice--regular in general. An alternative definition of exponential function, the $*$-exponential $exp_*$, was given: if $f$ is a slice--regular function, then $exp_*(f)$ is a slice--regular function as well. The study of a $*$-logarithm $log_*(f)$ of a slice--regular function $f$ becomes of great interest for basic reasons, and is performed in this paper. The main result shows that the existence of such a $log_*(f)$ depends only on the structure of the zero set of the vectorial part $f_v$ of the slice--regular function $f=f_0+f_v$, besides the topology of its domain of definition. We also show that, locally, every slice--regular nonvanishing function has a $*$-logarithm and, at the end, we present an example of a nonvanishing slice--regular function on a ball which does not admit a $*$-logarithm on that ball.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"534 ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41274853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Callias-type (or Dirac-Schr"odinger) operators associated to abstract semifinite spectral triples are introduced and their indices are computed in terms of an associated index pairing derived from the spectral triple. The result is then interpreted as an index theorem for a non-commutative analogue of spectral flow. Both even and odd spectral triples are considered, and both commutative and non-commutative examples are given.
{"title":"Callias-type operators associated to spectral triples","authors":"H. Schulz-Baldes, T. Stoiber","doi":"10.4171/jncg/505","DOIUrl":"https://doi.org/10.4171/jncg/505","url":null,"abstract":"Callias-type (or Dirac-Schr\"odinger) operators associated to abstract semifinite spectral triples are introduced and their indices are computed in terms of an associated index pairing derived from the spectral triple. The result is then interpreted as an index theorem for a non-commutative analogue of spectral flow. Both even and odd spectral triples are considered, and both commutative and non-commutative examples are given.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47531006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce a notion of strongly quasi-local algebras. They are defined for each discrete metric space with bounded geometry, and sit between the Roe algebra and the quasi-local algebra. We show that strongly quasi-local algebras are coarse invariants, hence encoding coarse geometric information of the underlying spaces. We prove that for a discrete metric space with bounded geometry which admits a coarse embedding into a Hilbert space, the inclusion of the Roe algebra into the strongly quasi-local algebra induces an isomorphism in $K$-theory.
{"title":"Strongly quasi-local algebras and their $K$-theories","authors":"HengDa Bao, Xiaoman Chen, Jiawen Zhang","doi":"10.4171/jncg/499","DOIUrl":"https://doi.org/10.4171/jncg/499","url":null,"abstract":"In this paper, we introduce a notion of strongly quasi-local algebras. They are defined for each discrete metric space with bounded geometry, and sit between the Roe algebra and the quasi-local algebra. We show that strongly quasi-local algebras are coarse invariants, hence encoding coarse geometric information of the underlying spaces. We prove that for a discrete metric space with bounded geometry which admits a coarse embedding into a Hilbert space, the inclusion of the Roe algebra into the strongly quasi-local algebra induces an isomorphism in $K$-theory.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46824655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). In a companion paper the authors described a noncommutative version of blowing down and, for example, gave a noncommutative analogue of Castelnuovo's classic theorem that lines of self-intersection (-1) on a smooth surface can be contracted. In this paper we will use these techniques to construct explicit birational transformations between various noncommutative surfaces containing an elliptic curve. Notably we show that Van den Bergh's quadrics can be obtained from the Sklyanin algebra by suitably blowing up and down, and we also provide a noncommutative analogue of the classical Cremona transform. This extends and amplifies earlier work of Presotto and Van den Bergh.
非交换代数几何中的一个主要开放问题是非交换投影曲面的分类(或者，更一般地说，是Gelfand-Kirillov维3的noetherian连通梯度域的分类)。在一篇合著的论文中，作者描述了一个非交换版本的吹落，例如，给出了Castelnuovo经典定理的一个非交换模拟，即光滑表面上的自交(-1)线可以被压缩。在本文中，我们将使用这些技术来构造包含椭圆曲线的各种非交换曲面之间的显式双分变换。值得注意的是，我们证明了Van den Bergh的二次曲面可以通过适当地放大和减小Sklyanin代数得到，并且我们还提供了经典Cremona变换的非交换模拟。这扩展和放大了普雷索托和范登伯格早期的工作。
{"title":"Ring-theoretic blowing down II: Birational transformations","authors":"D.Rogalski, S. J. Sierra, J. T. Stafford","doi":"10.4171/jncg/510","DOIUrl":"https://doi.org/10.4171/jncg/510","url":null,"abstract":"One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). In a companion paper the authors described a noncommutative version of blowing down and, for example, gave a noncommutative analogue of Castelnuovo's classic theorem that lines of self-intersection (-1) on a smooth surface can be contracted. In this paper we will use these techniques to construct explicit birational transformations between various noncommutative surfaces containing an elliptic curve. Notably we show that Van den Bergh's quadrics can be obtained from the Sklyanin algebra by suitably blowing up and down, and we also provide a noncommutative analogue of the classical Cremona transform. This extends and amplifies earlier work of Presotto and Van den Bergh.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"12 12","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41261296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deal with some questions regarding the notion of integral in the framework of Connes's noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman-Solomyak's perturbation theory. We also give a"soft proof"of Birman-Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassical Weyl's law for Schroedinger operators. This is an easy consequence of the Birman-Schwinger principle. We thus get a neat link between noncommutative geometry and semiclassical analysis.
{"title":"Connes' integration and Weyl's laws","authors":"Raphael Ponge","doi":"10.4171/jncg/509","DOIUrl":"https://doi.org/10.4171/jncg/509","url":null,"abstract":"This paper deal with some questions regarding the notion of integral in the framework of Connes's noncommutative geometry. First, we present a purely spectral theoretic construction of Connes' integral. This answers a question of Alain Connes. We also deal with the compatibility of Dixmier traces with Lebesgue's integral. This answers another question of Alain Connes. We further clarify the relationship of Connes' integration with Weyl's laws for compact operators and Birman-Solomyak's perturbation theory. We also give a\"soft proof\"of Birman-Solomyak's Weyl's law for negative order pseudodifferential operators on closed manifold. This Weyl's law yields a stronger form of Connes' trace theorem. Finally, we explain the relationship between Connes' integral and semiclassical Weyl's law for Schroedinger operators. This is an easy consequence of the Birman-Schwinger principle. We thus get a neat link between noncommutative geometry and semiclassical analysis.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43131927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Claudio Carmeli, Rita Fioresi, Veeravalli S. Varadarajan
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
给出了Lie超群和超代数的Harish-Chandra超模的酉性条件。
{"title":"Unitary Harish-Chandra representations of real supergroups","authors":"Claudio Carmeli, Rita Fioresi, Veeravalli S. Varadarajan","doi":"10.4171/jncg/496","DOIUrl":"https://doi.org/10.4171/jncg/496","url":null,"abstract":"We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44917756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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