Given an arbitrary countably generated rigid C*-tensor category, we construct a fully-faithful bi-involutive strong monoidal functor onto a subcategory of finitely generated projective bimodules over a simple, exact, separable, unital C*-algebra with unique trace. The C*-algebras involved are built from the category using the GJS-construction introduced in arXiv:0911.4728 and further studied in arXiv:1208.5505 and arXiv:1401.2486. Out of this category of Hilbert C*-bimodules, we construct a fully-faithful bi-involutive strong monoidal functor into the category of bi-finite spherical bimodules over an interpolated free group factor. The composite of these two functors recovers the functor constructed in arXiv:1208.5505
{"title":"Realizations of rigid C*-tensor categories as bimodules over GJS C*-algebras","authors":"Michael Hartglass, Roberto Hernández Palomares","doi":"10.1063/5.0015294","DOIUrl":"https://doi.org/10.1063/5.0015294","url":null,"abstract":"Given an arbitrary countably generated rigid C*-tensor category, we construct a fully-faithful bi-involutive strong monoidal functor onto a subcategory of finitely generated projective bimodules over a simple, exact, separable, unital C*-algebra with unique trace. The C*-algebras involved are built from the category using the GJS-construction introduced in arXiv:0911.4728 and further studied in arXiv:1208.5505 and arXiv:1401.2486. Out of this category of Hilbert C*-bimodules, we construct a fully-faithful bi-involutive strong monoidal functor into the category of bi-finite spherical bimodules over an interpolated free group factor. The composite of these two functors recovers the functor constructed in arXiv:1208.5505","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134488213","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 : 2020-04-21DOI: 10.1142/S0129055X21500215
Atibur Rahaman, Sutanu Roy
We construct a family of $q$ deformations of $E(2)$ groups for nonzero complex parameters $|q|<1$ as locally compact braided quantum groups over the circle group $mathbb{T}$ with respect to the unitary $R$-matrix $chicolonmathbb{Z}timesmathbb{Z}tomathbb{T}$ defined by $chi(m,n):=(zeta)^{mn}$, where $zeta:= q/bar{q}$. For real $0<|q|<1$, the deformation coincides with Woronowicz's $E_{q}(2)$ groups.
{"title":"Quantum E(2) groups for complex deformation parameters","authors":"Atibur Rahaman, Sutanu Roy","doi":"10.1142/S0129055X21500215","DOIUrl":"https://doi.org/10.1142/S0129055X21500215","url":null,"abstract":"We construct a family of $q$ deformations of $E(2)$ groups for nonzero complex parameters $|q|<1$ as locally compact braided quantum groups over the circle group $mathbb{T}$ with respect to the unitary $R$-matrix $chicolonmathbb{Z}timesmathbb{Z}tomathbb{T}$ defined by $chi(m,n):=(zeta)^{mn}$, where $zeta:= q/bar{q}$. For real $0<|q|<1$, the deformation coincides with Woronowicz's $E_{q}(2)$ groups.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116960369","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}
Let $mathfrak A$ be a type 1 subdiagonal algebra in a $sigma$-finite von Neumann algebra $mathcal M$ with respect to a faithful normal conditional expectation $Phi$. We give necessary and sufficient conditions for which $mathfrak A$ is maximal among the $sigma$-weakly closed subalgebras of $mathcal M$. In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of Arveson's finiteness problem in 1967 in type 1 case.
{"title":"Maximality and finiteness of type 1 subdiagonal algebras","authors":"Guoxing Ji","doi":"10.1090/proc/15287","DOIUrl":"https://doi.org/10.1090/proc/15287","url":null,"abstract":"Let $mathfrak A$ be a type 1 subdiagonal algebra in a $sigma$-finite von Neumann algebra $mathcal M$ with respect to a faithful normal conditional expectation $Phi$. We give necessary and sufficient conditions for which $mathfrak A$ is maximal among the $sigma$-weakly closed subalgebras of $mathcal M$. In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of Arveson's finiteness problem in 1967 in type 1 case.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125330620","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 : 2020-03-10DOI: 10.1216/RMJ.2020.50.1813
Eduardo Scarparo
Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $mathbb{Z}[frac{1}{pq}]rtimesmathbb{Z}^2$ admits a unique $mathrm{C}^*$-norm. The proof uses a characterization, due to Furstenberg, of closed $times p-$ and $times q-$invariant subsets of $mathbb{T}$.
{"title":"A torsion-free algebraically $mathrm{C}^*$-unique group","authors":"Eduardo Scarparo","doi":"10.1216/RMJ.2020.50.1813","DOIUrl":"https://doi.org/10.1216/RMJ.2020.50.1813","url":null,"abstract":"Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $mathbb{Z}[frac{1}{pq}]rtimesmathbb{Z}^2$ admits a unique $mathrm{C}^*$-norm. The proof uses a characterization, due to Furstenberg, of closed $times p-$ and $times q-$invariant subsets of $mathbb{T}$.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122012505","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 pushouts $A*_CB$ of residually finite-dimensional (RFD) $C^*$-algebras over central subalgebras are always residually finite-dimensional provided the fibers $A_p$ and $B_p$, $pin mathrm{spec}~C$ are RFD, recovering and generalizing results by Korchagin and Courtney-Shulman. This then allows us to prove that certain central pushouts of amenable groups have RFD group $C^*$-algebras. Along the way, we discuss the problem of when, given a central group embedding $Hle G$, the resulting $C^*$-algebra morphism is a continuous field: this is always the case for amenable $G$ but not in general.
{"title":"Residual finiteness for central pushouts","authors":"A. Chirvasitu","doi":"10.1090/proc/15368","DOIUrl":"https://doi.org/10.1090/proc/15368","url":null,"abstract":"We prove that pushouts $A*_CB$ of residually finite-dimensional (RFD) $C^*$-algebras over central subalgebras are always residually finite-dimensional provided the fibers $A_p$ and $B_p$, $pin mathrm{spec}~C$ are RFD, recovering and generalizing results by Korchagin and Courtney-Shulman. This then allows us to prove that certain central pushouts of amenable groups have RFD group $C^*$-algebras. Along the way, we discuss the problem of when, given a central group embedding $Hle G$, the resulting $C^*$-algebra morphism is a continuous field: this is always the case for amenable $G$ but not in general.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115551604","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 the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of Bader, Furman and Sauer. We apply this criterion to show that the categories constructed from totally disconnected groups by Arano and Vaes have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $Lambda
{"title":"L\u0000 2-Betti Numbers of C*-Tensor Categories Associated with Totally Disconnected Groups","authors":"Matthias Valvekens","doi":"10.1093/IMRN/RNAB066","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB066","url":null,"abstract":"We prove that the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of Bader, Furman and Sauer. We apply this criterion to show that the categories constructed from totally disconnected groups by Arano and Vaes have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $Lambda<Gamma$, with $Gamma$ acting on a type $mathrm{II}_1$ factor $P$ by outer automorphisms, we relate the cohomology theory of the quasi-regular inclusion $PrtimesLambdasubset PrtimesGamma$ to that of the Schlichting completion $G$ of $Lambda<Gamma$. If $Lambda<Gamma$ is unimodular, this correspondence allows us to prove that the $L^2$-Betti numbers of $PrtimesLambdasubset PrtimesGamma$ are equal to those of $G$.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124067094","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 a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups is inherited by the factors.
{"title":"Amenability and functoriality of right-LCM semigroup C*-algebras","authors":"Marcelo Laca, Boyu Li","doi":"10.1090/proc/15139","DOIUrl":"https://doi.org/10.1090/proc/15139","url":null,"abstract":"We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups is inherited by the factors.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130935964","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 paper, we introduce properties including groupoid comparison, pure infiniteness and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff '{e}tale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^*$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^*$-algebras arising from locally compact Hausdorff '{e}tale minimal topological principal groupoids. This generalizes the dichotomy obtained by B"{o}nicke-Li and Rainone-Sims. We also study the relation among our paradoxical comparison, $n$-filling property and locally contracting property appeared in the literature for locally compact Hausdorff '{e}tale groupoids.
{"title":"Purely Infinite Locally Compact Hausdorff étale Groupoids and Their C*-algebras","authors":"Xin Ma","doi":"10.1093/IMRN/RNAA360","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA360","url":null,"abstract":"In this paper, we introduce properties including groupoid comparison, pure infiniteness and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff '{e}tale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^*$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^*$-algebras arising from locally compact Hausdorff '{e}tale minimal topological principal groupoids. This generalizes the dichotomy obtained by B\"{o}nicke-Li and Rainone-Sims. We also study the relation among our paradoxical comparison, $n$-filling property and locally contracting property appeared in the literature for locally compact Hausdorff '{e}tale groupoids.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134318546","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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