In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~$ngeq 2$. This allows us to recover and improve earlier versions of this result in dimension $n=2$ and $n=4$ by Fathizadeh-Khalkhali. We also recover the Connes integration formula for flat noncommutative tori of McDonald-Sukochev-Zanin. As a further application we prove a curved version of this integration formula in terms of the Laplace-Beltrami operator defined by an arbitrary Riemannian metric. For the class of so-called self-compatible Riemannian metrics (including the conformally flat metrics of Connes-Tretkoff) this shows that Connes' noncommutative integral allows us to recover the Riemannian density. This exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras. As an application of these results, we setup a natural notion of scalar curvature for curved noncommutative tori.
{"title":"Connes’s trace theorem for curved noncommutative tori: Application to scalar curvature","authors":"Raphael Ponge","doi":"10.1063/5.0005052","DOIUrl":"https://doi.org/10.1063/5.0005052","url":null,"abstract":"In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~$ngeq 2$. This allows us to recover and improve earlier versions of this result in dimension $n=2$ and $n=4$ by Fathizadeh-Khalkhali. We also recover the Connes integration formula for flat noncommutative tori of McDonald-Sukochev-Zanin. As a further application we prove a curved version of this integration formula in terms of the Laplace-Beltrami operator defined by an arbitrary Riemannian metric. For the class of so-called self-compatible Riemannian metrics (including the conformally flat metrics of Connes-Tretkoff) this shows that Connes' noncommutative integral allows us to recover the Riemannian density. This exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras. As an application of these results, we setup a natural notion of scalar curvature for curved noncommutative tori.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125945507","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 establish uniqueness theorems for the noncommutative residue and the canonical trace on pseudodifferential operators on noncommutative tori of arbitrary dimension. The former is the unique trace up to constant multiple on integer order pseudodifferential operators. The latter is the unique trace up to constant multiple on non-integer order pseudodifferential operators. This improves previous uniqueness results by Fathizadeh-Khalkhali, Fathizadeh-Wong, and Levy-Neira-Paycha.
{"title":"Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results","authors":"Raphael Ponge","doi":"10.3842/sigma.2020.061","DOIUrl":"https://doi.org/10.3842/sigma.2020.061","url":null,"abstract":"In this paper we establish uniqueness theorems for the noncommutative residue and the canonical trace on pseudodifferential operators on noncommutative tori of arbitrary dimension. The former is the unique trace up to constant multiple on integer order pseudodifferential operators. The latter is the unique trace up to constant multiple on non-integer order pseudodifferential operators. This improves previous uniqueness results by Fathizadeh-Khalkhali, Fathizadeh-Wong, and Levy-Neira-Paycha.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124941592","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 : 2019-11-08DOI: 10.2422/2036-2145.201903_007
Valeriano Aiello, R. Conti, S. Rossi
A Fejer-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.
{"title":"A Fejér theorem for boundary quotients arising from algebraic dynamical systems","authors":"Valeriano Aiello, R. Conti, S. Rossi","doi":"10.2422/2036-2145.201903_007","DOIUrl":"https://doi.org/10.2422/2036-2145.201903_007","url":null,"abstract":"A Fejer-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127454192","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 construct some braided quantum groups over the circle group. These are analogous to the free orthogonal quantum groups and generalise the braided quantum SU(2) groups for complex deformation parameter. We describe their irreducible representations and fusion rules and study when they are monoidally equivalent.
{"title":"Braided Free Orthogonal Quantum Groups","authors":"R. Meyer, Sutanu Roy","doi":"10.1093/imrn/rnaa379","DOIUrl":"https://doi.org/10.1093/imrn/rnaa379","url":null,"abstract":"We construct some braided quantum groups over the circle group. These are analogous to the free orthogonal quantum groups and generalise the braided quantum SU(2) groups for complex deformation parameter. We describe their irreducible representations and fusion rules and study when they are monoidally equivalent.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131150660","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}
To every one-sided shift space $mathsf{X}$ we associate a cover $tilde{mathsf{X}}$, a groupoid $mathcal{G}_{mathsf{X}}$ and a $mathrm{C^*}$-algebra $mathcal{O}_{mathsf{X}}$. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer preserving) continuous orbit equivalence between $mathsf{X}$ and $mathsf{Y}$ in terms of isomorphism of $mathcal{G}_{mathsf{X}}$ and $mathcal{G}_{mathsf{Y}}$, and diagonal preserving $^*$-isomorphism of $mathcal{O}_{mathsf{X}}$ and $mathcal{O}_{mathsf{Y}}$. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces $Lambda_{mathsf{X}}$ and $Lambda_{mathsf{Y}}$ in terms of isomorphism of the stabilized groupoids $mathcal{G}_{mathsf{X}}times mathcal{R}$ and $mathcal{G}_{mathsf{Y}}times mathcal{R}$, and diagonal preserving $^*$-isomorphism of the stabilized $mathrm{C^*}$-algebras $mathcal{O}_{mathsf{X}}otimes mathbb{K}$ and $mathcal{O}_{mathsf{Y}}otimes mathbb{K}$. Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are essentially principal, we find that the pair $(mathcal{O}_{mathsf{X}}, C(mathsf{X}))$ remembers the continuous orbit equivalence class of $mathsf{X}$ while the pair $(mathcal{O}_{mathsf{X}}otimes mathbb{K}, C(mathsf{X})otimes c_0)$ remembers the flow equivalence class of $Lambda_{mathsf{X}}$. In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.
{"title":"𝐶*-algebras, groupoids and covers of shift spaces","authors":"K. Brix, T. M. Carlsen","doi":"10.1090/btran/53","DOIUrl":"https://doi.org/10.1090/btran/53","url":null,"abstract":"To every one-sided shift space $mathsf{X}$ we associate a cover $tilde{mathsf{X}}$, a groupoid $mathcal{G}_{mathsf{X}}$ and a $mathrm{C^*}$-algebra $mathcal{O}_{mathsf{X}}$. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer preserving) continuous orbit equivalence between $mathsf{X}$ and $mathsf{Y}$ in terms of isomorphism of $mathcal{G}_{mathsf{X}}$ and $mathcal{G}_{mathsf{Y}}$, and diagonal preserving $^*$-isomorphism of $mathcal{O}_{mathsf{X}}$ and $mathcal{O}_{mathsf{Y}}$. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces $Lambda_{mathsf{X}}$ and $Lambda_{mathsf{Y}}$ in terms of isomorphism of the stabilized groupoids $mathcal{G}_{mathsf{X}}times mathcal{R}$ and $mathcal{G}_{mathsf{Y}}times mathcal{R}$, and diagonal preserving $^*$-isomorphism of the stabilized $mathrm{C^*}$-algebras $mathcal{O}_{mathsf{X}}otimes mathbb{K}$ and $mathcal{O}_{mathsf{Y}}otimes mathbb{K}$. Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are essentially principal, we find that the pair $(mathcal{O}_{mathsf{X}}, C(mathsf{X}))$ remembers the continuous orbit equivalence class of $mathsf{X}$ while the pair $(mathcal{O}_{mathsf{X}}otimes mathbb{K}, C(mathsf{X})otimes c_0)$ remembers the flow equivalence class of $Lambda_{mathsf{X}}$. In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"95 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133847090","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}
Rolando de Santiago, Ben Hayes, D. Hoff, Thomas Sinclair
In the past two decades, Sorin Popa's breakthrough deformation/rigidity theory has produced remarkable rigidity results for von Neumann algebras $M$ which can be deformed inside a larger algebra $widetilde M supseteq M$ by an action $alpha: mathbb{R} to {rm Aut}(widetilde M)$, while simultaneously containing subalgebras $Q$ {it rigid} with respect to that deformation, that is, such that $alpha_t to {rm id}$ uniformly on the unit ball of $Q$ as $t to 0$. However, it has remained unclear how to exploit the interplay between distinct rigid subalgebras not in specified relative position. We show that in fact, any diffuse subalgebra which is rigid with respect to a mixing s-malleable deformation is contained in a subalgebra which is uniquely maximal with respect to being rigid. In particular, the algebra generated by any family of rigid subalgebras that intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members was the motivation for this work, showing for example that if $G$ is a countable group with $beta^{1}_{(2)}(G) > 0$, then $L(G)$ cannot be generated by two property $(T)$ subalgebras with diffuse intersection; however, the result is most striking when the family is infinite.
在过去的二十年里，Sorin Popa的突破性变形/刚性理论已经对von Neumann代数$M$产生了显著的刚性结果，这些代数可以在一个更大的代数$widetilde M supseteq M$中通过一个作用$alpha: mathbb{R} to {rm Aut}(widetilde M)$进行变形，同时包含子代数$Q$相对于该变形是{it刚性}的，即$alpha_t to {rm id}$在$Q$的单位球上均匀地为$t to 0$。然而，它仍然不清楚如何利用不同的刚性子代数之间的相互作用，而不是在指定的相对位置。我们证明了事实上，任何对混合s-可塑变形具有刚性的漫射子代数都包含在一个对刚性具有唯一极大的子代数中。特别地，由任意一组扩散相交的刚性子代数所生成的代数，就该变形而言，本身必须是刚性的。这个族有两个成员的情况是这项工作的动机，例如，如果$G$是一个具有$beta^{1}_{(2)}(G) > 0$的可数群，那么$L(G)$不能由两个具有漫射相交的性质$(T)$子代数生成;然而，当家庭是无限的时候，结果是最惊人的。
{"title":"Maximal Rigid Subalgebras of Deformations and $L^2$ Cohomology, II","authors":"Rolando de Santiago, Ben Hayes, D. Hoff, Thomas Sinclair","doi":"10.14288/1.0389705","DOIUrl":"https://doi.org/10.14288/1.0389705","url":null,"abstract":"In the past two decades, Sorin Popa's breakthrough deformation/rigidity theory has produced remarkable rigidity results for von Neumann algebras $M$ which can be deformed inside a larger algebra $widetilde M supseteq M$ by an action $alpha: mathbb{R} to {rm Aut}(widetilde M)$, while simultaneously containing subalgebras $Q$ {it rigid} with respect to that deformation, that is, such that $alpha_t to {rm id}$ uniformly on the unit ball of $Q$ as $t to 0$. However, it has remained unclear how to exploit the interplay between distinct rigid subalgebras not in specified relative position. \u0000We show that in fact, any diffuse subalgebra which is rigid with respect to a mixing s-malleable deformation is contained in a subalgebra which is uniquely maximal with respect to being rigid. In particular, the algebra generated by any family of rigid subalgebras that intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members was the motivation for this work, showing for example that if $G$ is a countable group with $beta^{1}_{(2)}(G) > 0$, then $L(G)$ cannot be generated by two property $(T)$ subalgebras with diffuse intersection; however, the result is most striking when the family is infinite.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1009 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123098056","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 : 2019-09-04DOI: 10.1142/s0129167x20500469
Li Gao, M. Junge, Nicholas Laracuente
We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2le ple infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy with respect to a subalgebra. These relative entropy generalizes subfactors index and has application in estimating decoherence time of quantum Markov semigroup.
{"title":"Relative entropy for von Neumann subalgebras","authors":"Li Gao, M. Junge, Nicholas Laracuente","doi":"10.1142/s0129167x20500469","DOIUrl":"https://doi.org/10.1142/s0129167x20500469","url":null,"abstract":"We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2le ple infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy with respect to a subalgebra. These relative entropy generalizes subfactors index and has application in estimating decoherence time of quantum Markov semigroup.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125761825","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 consider a set $SPG(mathcal{A})$ of pure split states on a quantum spin chain $mathcal{A}$ which are invariant under the on-site action $tau$ of a finite group $G$. For each element $omega$ in $SPG(mathcal{A})$ we can associate a second cohomology class $c_{omega,R}$of $G$. We consider a classification of $SPG(mathcal{A})$ whose criterion is given as follows: $omega_{0}$ and $omega_{1}$ in $SPG(mathcal{A})$ are equivalent if there are automorphisms $Xi_{R}$, $Xi_L$ on $mathcal{A}_{R}$, $mathcal{A}_{L}$ (right and left half infinite chains) preserving the symmetry $tau$, such that $omega_{1}$ and $omega_{0}circ( Xi_{L}otimes Xi_{R})$ are quasi-equivalent. It means that we can move $omega_{0}$ close to $omega_{1}$ without changing the entanglement nor breaking the symmetry. We show that the second cohomology class $c_{omega,R}$ is the complete invariant of this classification.
{"title":"A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries","authors":"Y. Ogata","doi":"10.1090/BTRAN/51","DOIUrl":"https://doi.org/10.1090/BTRAN/51","url":null,"abstract":"We consider a set $SPG(mathcal{A})$ of pure split states on a quantum spin chain $mathcal{A}$ which are invariant under the on-site action $tau$ of a finite group $G$. For each element $omega$ in $SPG(mathcal{A})$ we can associate a second cohomology class $c_{omega,R}$of $G$. We consider a classification of $SPG(mathcal{A})$ whose criterion is given as follows: $omega_{0}$ and $omega_{1}$ in $SPG(mathcal{A})$ are equivalent if there are automorphisms $Xi_{R}$, $Xi_L$ on $mathcal{A}_{R}$, $mathcal{A}_{L}$ (right and left half infinite chains) preserving the symmetry $tau$, such that $omega_{1}$ and $omega_{0}circ( Xi_{L}otimes Xi_{R})$ are quasi-equivalent. It means that we can move $omega_{0}$ close to $omega_{1}$ without changing the entanglement nor breaking the symmetry. We show that the second cohomology class $c_{omega,R}$ is the complete invariant of this classification.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122479613","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 : 2019-07-24DOI: 10.1016/j.jfa.2021.109027
K. Dykema, Amudhan Krishnaswamy-Usha
{"title":"Angles between Haagerup–Schultz projections and spectrality of operators","authors":"K. Dykema, Amudhan Krishnaswamy-Usha","doi":"10.1016/j.jfa.2021.109027","DOIUrl":"https://doi.org/10.1016/j.jfa.2021.109027","url":null,"abstract":"","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120243611","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}
Given a group cocycle on a finitely aligned left cancellative small category (LCSC) we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSC's, and to construct coactions of groups on Exel-Pardo algebras. Finally we discuss the universal group of a small category and connectedness of skew product categories.
{"title":"Skew products of finitely aligned left cancellative small categories and Cuntz-Krieger algebras","authors":"Erik B'edos, S. Kaliszewski, John Quigg","doi":"10.17879/59019527597","DOIUrl":"https://doi.org/10.17879/59019527597","url":null,"abstract":"Given a group cocycle on a finitely aligned left cancellative small category (LCSC) we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSC's, and to construct coactions of groups on Exel-Pardo algebras. Finally we discuss the universal group of a small category and connectedness of skew product categories.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116019213","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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