We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers, Restorff, Ruiz and S{o}rensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type, and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).
{"title":"Decidability of flow equivalence and isomorphism problems for graph $C^*$-algebras and quiver representations","authors":"Mike Boyle, B. Steinberg","doi":"10.1090/proc/15148","DOIUrl":"https://doi.org/10.1090/proc/15148","url":null,"abstract":"We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers, Restorff, Ruiz and S{o}rensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type, and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"9 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":"129247820","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 show the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type III$_1$ with finite index, or an irreducible discrete inclusion whose small algebra is an injective factor of type III$_1$.
{"title":"On the Relative Bicentralizer Flows and the Relative Flow of Weights of Inclusions of Factors of Type III$_1$","authors":"Toshihiko Masuda","doi":"10.4171/prims/56-2-4","DOIUrl":"https://doi.org/10.4171/prims/56-2-4","url":null,"abstract":"We show the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type III$_1$ with finite index, or an irreducible discrete inclusion whose small algebra is an injective factor of type III$_1$.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129556983","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 paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics.
{"title":"Asymptotic free independence and entry permutations for Gaussian random matrices","authors":"M. Popa","doi":"10.1090/proc/15783","DOIUrl":"https://doi.org/10.1090/proc/15783","url":null,"abstract":"The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129958414","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 one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $mathbb{R}$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms. Furthermore, we obtain analogous results for the setting of decreasing Loewner chains in the unit disk, which correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.
{"title":"Monotone increment processes, classical Markov processes, and Loewner chains","authors":"U. Franz, Takahiro Hasebe, Sebastian Schleißinger","doi":"10.4064/dm808-1-2020","DOIUrl":"https://doi.org/10.4064/dm808-1-2020","url":null,"abstract":"We prove one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $mathbb{R}$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms. \u0000Furthermore, we obtain analogous results for the setting of decreasing Loewner chains in the unit disk, which correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125964053","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 : 2018-10-25DOI: 10.1017/1017/S0017089519000053
James Gabe, Efren Ruiz
The semigroups of unital extensions of separable $C^ast$-algebras come in two flavours: a strong and a weak version. By the unital $mathrm{Ext}$-groups, we mean the groups of invertible elements in these semigroups. We use the unital $mathrm{Ext}$-groups to obtain $K$-theoretic classification of both unital and non-unital extensions of $C^ast$-algebras, and in particular we obtain a complete $K$-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.
{"title":"The unital Ext-groups and classification of $C^ast$-algebras","authors":"James Gabe, Efren Ruiz","doi":"10.1017/1017/S0017089519000053","DOIUrl":"https://doi.org/10.1017/1017/S0017089519000053","url":null,"abstract":"The semigroups of unital extensions of separable $C^ast$-algebras come in two flavours: a strong and a weak version. By the unital $mathrm{Ext}$-groups, we mean the groups of invertible elements in these semigroups. We use the unital $mathrm{Ext}$-groups to obtain $K$-theoretic classification of both unital and non-unital extensions of $C^ast$-algebras, and in particular we obtain a complete $K$-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121475943","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 2015, Raum and Weber gave a definition of group-theoretical quantum groups, a class of compact matrix quantum groups with a certain presentation as semi-direct product quantum groups, and studied the case of easy quantum groups. In this article we determine the intertwiner spaces of non-easy group-theoretical quantum groups. We generalise group-theoretical categories of partitions and use a fiber functor to map partitions to linear maps which is slightly different from the one for easy quantum groups. We show that this construction provides the intertwiner spaces of group-theoretical quantum groups in general.
{"title":"The intertwiner spaces of non-easy group-theoretical quantum groups","authors":"Laura Maassen","doi":"10.4171/jncg/384","DOIUrl":"https://doi.org/10.4171/jncg/384","url":null,"abstract":"In 2015, Raum and Weber gave a definition of group-theoretical quantum groups, a class of compact matrix quantum groups with a certain presentation as semi-direct product quantum groups, and studied the case of easy quantum groups. In this article we determine the intertwiner spaces of non-easy group-theoretical quantum groups. We generalise group-theoretical categories of partitions and use a fiber functor to map partitions to linear maps which is slightly different from the one for easy quantum groups. We show that this construction provides the intertwiner spaces of group-theoretical quantum groups in general.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122147291","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 study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $Bsubseteq A$ of $mathrm{C}^*$-algebras is isomorphic to the canonical inclusion of $ell^infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for `Roe Cartans' inside uniform Roe algebras up to automorphism when $X$ coarsely embeds into Hilbert space, and up to inner automorphism when $X$ has property A.
{"title":"Cartan subalgebras in uniform Roe algebras","authors":"Stuart White, R. Willett","doi":"10.4171/ggd/570","DOIUrl":"https://doi.org/10.4171/ggd/570","url":null,"abstract":"In this paper we study structural and uniqueness questions for Cartan subalgebras of uniform Roe algebras. We characterise when an inclusion $Bsubseteq A$ of $mathrm{C}^*$-algebras is isomorphic to the canonical inclusion of $ell^infty(X)$ inside a uniform Roe algebra $C^*_u(X)$ associated to a metric space of bounded geometry. We obtain uniqueness results for `Roe Cartans' inside uniform Roe algebras up to automorphism when $X$ coarsely embeds into Hilbert space, and up to inner automorphism when $X$ has property A.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128629310","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 : 2018-07-19DOI: 10.4007/annals.2020.192.2.1
Christopher Schafhauser
It is shown that if A is a separable, exact C*-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with unique trace. Modulo the UCT, this provides an abstract characterization of C*-subalgebras of simple, unital AF-algebras. As a consequence, for a countable, discrete, amenable group G acting on a second countable, locally compact, Hausdorff space X, C_0(X) rtimes_r G embeds into a simple, unital AF-algebra if, and only if, X admits a faithful, invariant, Borel, probability measure. Also, for any countable, discrete, amenable group G, the reduced group C*-algebra C*_r(G) admits a trace-preserving embedding into the universal UHF-algebra.
{"title":"Subalgebras of simple AF-algebras","authors":"Christopher Schafhauser","doi":"10.4007/annals.2020.192.2.1","DOIUrl":"https://doi.org/10.4007/annals.2020.192.2.1","url":null,"abstract":"It is shown that if A is a separable, exact C*-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with unique trace. Modulo the UCT, this provides an abstract characterization of C*-subalgebras of simple, unital AF-algebras. \u0000As a consequence, for a countable, discrete, amenable group G acting on a second countable, locally compact, Hausdorff space X, C_0(X) rtimes_r G embeds into a simple, unital AF-algebra if, and only if, X admits a faithful, invariant, Borel, probability measure. Also, for any countable, discrete, amenable group G, the reduced group C*-algebra C*_r(G) admits a trace-preserving embedding into the universal UHF-algebra.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134216335","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 compute the homology of the groupoid associated to the Katsura algebras, and show that they capture the $K$-theory of the $C^*$-algebras, and hence satisfying the (HK) conjecture posted by Matui. Moreover, we show that several classifiable simple $C^*$-algebras are groupoid $C^*$-algebras of this class.
{"title":"The homology of the Katsura–Exel–Pardo groupoid","authors":"E. Ortega","doi":"10.4171/jncg/382","DOIUrl":"https://doi.org/10.4171/jncg/382","url":null,"abstract":"We compute the homology of the groupoid associated to the Katsura algebras, and show that they capture the $K$-theory of the $C^*$-algebras, and hence satisfying the (HK) conjecture posted by Matui. Moreover, we show that several classifiable simple $C^*$-algebras are groupoid $C^*$-algebras of this class.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126367538","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 : 2018-06-07DOI: 10.22072/wala.2018.79871.1153
M. Rossafi, S. Kabbaj
In this paper, a new notion of frames is introduced: $ast$-operator frame as generalization of $ast$-frames in Hilbert $C^{ast}$-modules introduced by A. Alijani and M. A. Dehghan cite{Ali} and we establish some results.
{"title":"$ast$-operator frame for $End_{mathcal{A}}^{ast}(mathcal{H})$","authors":"M. Rossafi, S. Kabbaj","doi":"10.22072/wala.2018.79871.1153","DOIUrl":"https://doi.org/10.22072/wala.2018.79871.1153","url":null,"abstract":"In this paper, a new notion of frames is introduced: $ast$-operator frame as generalization of $ast$-frames in Hilbert $C^{ast}$-modules introduced by A. Alijani and M. A. Dehghan cite{Ali} and we establish some results.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116533010","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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