Pub Date : 2019-04-30DOI: 10.13140/RG.2.2.25496.65289
P. Ivankov
This research is devoted to the noncommutative generalization of topological coverings. Otherwise since topological coverings are related to the set of geometric constructions one can obtain noncommutative generalizations of these constructions. Here the generalizations of the universal covering space, fundamental group, covering of the Riemann manifolds, flat connections are explained. The theory gives pure algebraic proof well known results of the topology and the differential geometry. Besides there are applications of the theory to noncommutative $C^*$-algebras and this theme is also discussed here.
{"title":"Noncommutative Geometry of Quantized Coverings","authors":"P. Ivankov","doi":"10.13140/RG.2.2.25496.65289","DOIUrl":"https://doi.org/10.13140/RG.2.2.25496.65289","url":null,"abstract":"This research is devoted to the noncommutative generalization of topological coverings. Otherwise since topological coverings are related to the set of geometric constructions one can obtain noncommutative generalizations of these constructions. Here the generalizations of the universal covering space, fundamental group, covering of the Riemann manifolds, flat connections are explained. The theory gives pure algebraic proof well known results of the topology and the differential geometry. Besides there are applications of the theory to noncommutative $C^*$-algebras and this theme is also discussed here.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125647681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we propose a $p$-adic analogue of complex Hilbert space and consider generalizations of some well-known theorems from functional analysis and the basic study of operators on Hilbert spaces. We compute the $K$-theory of the analogue of the algebra of compact operators and the algebra of all bounded operators. This article contains a survey on results from the thesis of the first author.
{"title":"Aspects of $p$-adic operator algebras","authors":"A. Claussnitzer, A. Thom","doi":"10.17879/90169651061","DOIUrl":"https://doi.org/10.17879/90169651061","url":null,"abstract":"In this article, we propose a $p$-adic analogue of complex Hilbert space and consider generalizations of some well-known theorems from functional analysis and the basic study of operators on Hilbert spaces. We compute the $K$-theory of the analogue of the algebra of compact operators and the algebra of all bounded operators. This article contains a survey on results from the thesis of the first author.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"227 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115005283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Hamhalter, Ondvrej F. K. Kalenda, A. M. Peralta, H. Pfitzner
We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $psiin E^*$ satisfying $$|T(x)| leq K , |T| , |x|_{psi},$$ for all $xin E$. Applying this result we show that, given $G > 8 (1+2sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product of two JB$^*$-triples $E$ and $B$, there exist norm-one functionals $varphiin E^{*}$ and $psiin B^{*}$ satisfying $$|V(x,y)| leq G |V| , |x|_{varphi} , |y|_{psi}$$ for all $(x,y)in E times B$. These results prove a conjecture pursued during almost twenty years.
我们证明，给定一个常数$K> 2$和一个有界线性算子$T$，从一个JB $^*$ -三元组$E$到一个复希尔伯特空间$H$，存在一个范数一泛函$psiin E^*$，对所有$xin E$满足$$|T(x)| leq K , |T| , |x|_{psi},$$。应用这一结果，我们证明了在两个JB - $^*$ -三元组$E$和$B$的笛卡尔积上，给定$G > 8 (1+2sqrt{3})$和有界双线性形式$V$，存在对所有$(x,y)in E times B$满足$$|V(x,y)| leq G |V| , |x|_{varphi} , |y|_{psi}$$的范数一泛函$varphiin E^{*}$和$psiin B^{*}$。这些结果证明了一个研究了近二十年的猜想。
{"title":"Grothendieck’s inequalities for JB$^*$-triples: Proof of the Barton–Friedman conjecture","authors":"J. Hamhalter, Ondvrej F. K. Kalenda, A. M. Peralta, H. Pfitzner","doi":"10.1090/tran/8227","DOIUrl":"https://doi.org/10.1090/tran/8227","url":null,"abstract":"We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $psiin E^*$ satisfying $$|T(x)| leq K , |T| , |x|_{psi},$$ for all $xin E$. Applying this result we show that, given $G > 8 (1+2sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product of two JB$^*$-triples $E$ and $B$, there exist norm-one functionals $varphiin E^{*}$ and $psiin B^{*}$ satisfying $$|V(x,y)| leq G |V| , |x|_{varphi} , |y|_{psi}$$ for all $(x,y)in E times B$. These results prove a conjecture pursued during almost twenty years.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130055632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.
{"title":"Smale space $C^*$-algebras have nonzero projections","authors":"R. Deeley, M. Goffeng, A. Yashinski","doi":"10.1090/proc/14837","DOIUrl":"https://doi.org/10.1090/proc/14837","url":null,"abstract":"The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114723706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that a locally compact group has the approximation property (AP), introduced by Haagerup-Kraus, if and only if a non-commutative Fej'{e}r theorem holds for the associated $C^*$- or von Neumann crossed products. As applications, we answer three open problems in the literature. Specifically, we show that any locally compact group with the AP is exact. This generalizes a result by Haagerup-Kraus, and answers a problem raised by Li. We also answer a question of B'{e}dos-Conti on the Fej'{e}r property of discrete $C^*$-dynamical systems, as well as a question by Anoussis-Katavolos-Todorov for all locally compact groups with the AP. In our approach, which relies on operator space techniques, we develop a notion of Fubini crossed product for locally compact groups, and a dynamical version of the AP for actions associated with $C^*$- or $W^*$-dynamical systems.
{"title":"A Non-commutative Fejér Theorem for Crossed Products, the Approximation Property, and Applications","authors":"Jason Crann, M. Neufang","doi":"10.1093/imrn/rnaa221","DOIUrl":"https://doi.org/10.1093/imrn/rnaa221","url":null,"abstract":"We prove that a locally compact group has the approximation property (AP), introduced by Haagerup-Kraus, if and only if a non-commutative Fej'{e}r theorem holds for the associated $C^*$- or von Neumann crossed products. As applications, we answer three open problems in the literature. Specifically, we show that any locally compact group with the AP is exact. This generalizes a result by Haagerup-Kraus, and answers a problem raised by Li. We also answer a question of B'{e}dos-Conti on the Fej'{e}r property of discrete $C^*$-dynamical systems, as well as a question by Anoussis-Katavolos-Todorov for all locally compact groups with the AP. In our approach, which relies on operator space techniques, we develop a notion of Fubini crossed product for locally compact groups, and a dynamical version of the AP for actions associated with $C^*$- or $W^*$-dynamical systems.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116804729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital $*$-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense $*$-subalgebra of the unitisation of the universal C$^*$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups.
{"title":"Generating Functionals for Locally Compact Quantum Groups","authors":"Adam G. Skalski, Ami Viselter","doi":"10.1093/imrn/rnz387","DOIUrl":"https://doi.org/10.1093/imrn/rnz387","url":null,"abstract":"Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital $*$-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense $*$-subalgebra of the unitisation of the universal C$^*$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116248663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend a theorem of Haagerup and Kraus in the C*-algebra context: for a locally compact group with the approximation property (AP), the reduced C*-crossed product construction preserves the strong operator approximation property (SOAP). In particular their reduced group C*-algebras have the SOAP. Our method also solves another open problem: the AP implies exactness for general locally compact groups.
{"title":"The approximation property and exactness of locally compact groups","authors":"Yuhei Suzuki","doi":"10.2969/jmsj/83368336","DOIUrl":"https://doi.org/10.2969/jmsj/83368336","url":null,"abstract":"We extend a theorem of Haagerup and Kraus in the C*-algebra context: for a locally compact group with the approximation property (AP), the reduced C*-crossed product construction preserves the strong operator approximation property (SOAP). In particular their reduced group C*-algebras have the SOAP. Our method also solves another open problem: the AP implies exactness for general locally compact groups.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126067600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-03DOI: 10.1142/S1793525321500035
W. Lueck
We consider the problem whether for a group G there exists a constant Lambda(G) > 1 such that for any (r,s)-matrix A over the integral group ring ZG the Fuglede-Kadison determinant of the G-equivariant bounded operator from L^2(G)^r to L^2(G)^s given by right multiplication with A is either one or greater or equal to Lambda(G). If G is the infinite cyclic group and we consider only r = s = 1, this is precisely Lehmer's problem.
考虑对于群G是否存在一个常数λ (G) > 1，使得对于整群环ZG上的任意(r,s)-矩阵a，由与a右乘给出的从L^2(G)^r到L^2(G)^s的G-等变有界算子的Fuglede-Kadison行列式等于或大于等于λ (G)。如果G是无限循环群，我们只考虑r = s = 1，这就是Lehmer问题。
{"title":"Lehmer’s problem for arbitrary groups","authors":"W. Lueck","doi":"10.1142/S1793525321500035","DOIUrl":"https://doi.org/10.1142/S1793525321500035","url":null,"abstract":"We consider the problem whether for a group G there exists a constant Lambda(G) > 1 such that for any (r,s)-matrix A over the integral group ring ZG the Fuglede-Kadison determinant of the G-equivariant bounded operator from L^2(G)^r to L^2(G)^s given by right multiplication with A is either one or greater or equal to Lambda(G). If G is the infinite cyclic group and we consider only r = s = 1, this is precisely Lehmer's problem.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"161 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133119169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary of this main orbit. As examples, we discuss various classes of magnetic pseudo-differential operators on nilpotent groups. We also prove localization and non-propagation properties associated to suitable parts of the essential spectrum. These are applied to twisted groupoids having a totally intransitive groupoid restriction at the boundary.
{"title":"Spectral Theory in a Twisted Groupoid Setting: Spectral Decompositions, Localization and Fredholmness","authors":"M. Măntoiu, V. Nistor","doi":"10.17879/32109552889","DOIUrl":"https://doi.org/10.17879/32109552889","url":null,"abstract":"We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary of this main orbit. As examples, we discuss various classes of magnetic pseudo-differential operators on nilpotent groups. We also prove localization and non-propagation properties associated to suitable parts of the essential spectrum. These are applied to twisted groupoids having a totally intransitive groupoid restriction at the boundary.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"235 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132776071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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