We show that if every element a vector subspace of a C*-algebra can be averaged to zero by means of unitary mixing operators, then all the elements of the subspace can be simultaneously averaged to zero by a net of unitary mixing operators. Moreover, such subspaces admit a simple description in terms of commutators and kernels of states on the C*-algebra. We apply this result to center-valued expectations in C*-algebras with the Dixmier property.
{"title":"Simultaneous averaging to zero by unitary mixing operators","authors":"Abhinav Chand, L. Robert, Arindam Sutradhar","doi":"10.1090/PROC/15495","DOIUrl":"https://doi.org/10.1090/PROC/15495","url":null,"abstract":"We show that if every element a vector subspace of a C*-algebra can be averaged to zero by means of unitary mixing operators, then all the elements of the subspace can be simultaneously averaged to zero by a net of unitary mixing operators. Moreover, such subspaces admit a simple description in terms of commutators and kernels of states on the C*-algebra. We apply this result to center-valued expectations in C*-algebras with the Dixmier property.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114926826","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 number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples $(X_1,cdots,X_d)$ of traceless self-adjoint operators in the $ntimes n$ matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: $2le dle n^2-3$. Moreover, the automorphism group is generically abelian in the larger parameter range $1le dle n^2-2$. This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of $X_i$'s (mimicking the ErdH{o}s-R'{e}nyi $G(n,p)$ model) has trivial/abelian automorphism group almost surely.
我们证明了一般量子图(在Duan-Severini-Winter / Weaver的工作中通过算子系统定义)具有很少的对称性:对于$n n$矩阵代数中无迹自伴随算子的元组$(X_1，cdots,X_d)$的zariski稠密开集，对应的算子系统在参数$2le dle n^2-3$的最大可能范围内具有平凡自同构群。此外，自同构群在较大的参数范围$1le dle n^2-2$内是一般的阿贝尔群。这就意味着对于这些相应的参数，建立在$X_i$ s的GUE综上的相应的随机量子图模型(模仿ErdH{o}s r {e}nyi $G(n,p)$模型)几乎肯定具有平凡/阿贝尔自同构群。
{"title":"Random quantum graphs","authors":"A. Chirvasitu, Mateusz Wasilewski","doi":"10.1090/tran/8584","DOIUrl":"https://doi.org/10.1090/tran/8584","url":null,"abstract":"We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples $(X_1,cdots,X_d)$ of traceless self-adjoint operators in the $ntimes n$ matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: $2le dle n^2-3$. Moreover, the automorphism group is generically abelian in the larger parameter range $1le dle n^2-2$. This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of $X_i$'s (mimicking the ErdH{o}s-R'{e}nyi $G(n,p)$ model) has trivial/abelian automorphism group almost surely.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131206702","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 prove a $K$-homology index theorem for the Toeplitz operators obtained from the multishifts of the Bergman space on several classes of egg-like domains. This generalizes our theorem with Douglas and Yu on the unit ball.
{"title":"An index theorem for quotients of Bergman spaces on egg domains","authors":"M. Jabbari, Xiang Tang","doi":"10.2140/akt.2021.6.357","DOIUrl":"https://doi.org/10.2140/akt.2021.6.357","url":null,"abstract":"In this paper we prove a $K$-homology index theorem for the Toeplitz operators obtained from the multishifts of the Bergman space on several classes of egg-like domains. This generalizes our theorem with Douglas and Yu on the unit ball.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130579312","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 : 2020-09-12DOI: 10.1142/s0129167x21500051
A. Bearden, Jason Crann
We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras $A$. We prove a number of general results---for example, a characterization of the $A$-WEP in terms of an appropriate $A$-injective envelope, and also a characterization of those $A$ for which $A$-WEP implies WEP. In the case of $A=L^1(G)$, we recover the $G$-WEP for $G$-$C^*$-algebras in recent work of Buss--Echterhoff--Willett. When $A=A(G)$, we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related in the setting of dynamical systems, where we show that a $W^*$-dynamical system $(M,G,alpha)$ with $M$ injective is amenable if and only if $M$ is $L^1(G)$-injective if and only if the crossed product $Gbar{ltimes}M$ is $A(G)$-injective. Analogously, we show that a $C^*$-dynamical system $(A,G,alpha)$ with $A$ nuclear and $G$ exact is amenable if and only if $A$ has the $L^1(G)$-WEP if and only if the reduced crossed product $Gltimes A$ has the $A(G)$-WEP.
{"title":"A weak expectation property for operator modules, injectivity and amenable actions","authors":"A. Bearden, Jason Crann","doi":"10.1142/s0129167x21500051","DOIUrl":"https://doi.org/10.1142/s0129167x21500051","url":null,"abstract":"We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras $A$. We prove a number of general results---for example, a characterization of the $A$-WEP in terms of an appropriate $A$-injective envelope, and also a characterization of those $A$ for which $A$-WEP implies WEP. In the case of $A=L^1(G)$, we recover the $G$-WEP for $G$-$C^*$-algebras in recent work of Buss--Echterhoff--Willett. When $A=A(G)$, we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related in the setting of dynamical systems, where we show that a $W^*$-dynamical system $(M,G,alpha)$ with $M$ injective is amenable if and only if $M$ is $L^1(G)$-injective if and only if the crossed product $Gbar{ltimes}M$ is $A(G)$-injective. Analogously, we show that a $C^*$-dynamical system $(A,G,alpha)$ with $A$ nuclear and $G$ exact is amenable if and only if $A$ has the $L^1(G)$-WEP if and only if the reduced crossed product $Gltimes A$ has the $A(G)$-WEP.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132591270","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 give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $Gamma$ implies that $Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.
{"title":"On the Baum-Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture","authors":"Yuki Arano, Adam G. Skalski","doi":"10.1090/PROC/15598","DOIUrl":"https://doi.org/10.1090/PROC/15598","url":null,"abstract":"We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $Gamma$ implies that $Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130949838","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 implications among the conditions in the title for an inclusion of a C*-algebra A in a C*-algebra B, and we also relate this to several other properties in case B is a crossed product for an action of a group, inverse semigroup or an etale groupoid on A. We show that an aperiodic C*-inclusion has a unique pseudo-expectation. If, in addition, the unique pseudo-expectation is faithful, then A supports B in the sense of the Cuntz preorder. The almost extension property implies aperiodicity, and the converse holds if B is separable. A crossed product inclusion has the almost extension property if and only if the dual groupoid of the action is topologically principal. Topologically free actions are always aperiodic. If A is separable or of Type I, then topological freeness, aperiodicity and having a unique pseudo-expectation are equivalent to the condition that A detects ideals in all intermediate C*-algebras. If, in addition, B is separable, then all these conditions are equivalent to the almost extension property.
{"title":"Aperiodicity: The Almost Extension Property and Uniqueness of Pseudo-Expectations","authors":"B. Kwa'sniewski, R. Meyer","doi":"10.1093/imrn/rnab098","DOIUrl":"https://doi.org/10.1093/imrn/rnab098","url":null,"abstract":"We prove implications among the conditions in the title for an inclusion of a C*-algebra A in a C*-algebra B, and we also relate this to several other properties in case B is a crossed product for an action of a group, inverse semigroup or an etale groupoid on A. We show that an aperiodic C*-inclusion has a unique pseudo-expectation. If, in addition, the unique pseudo-expectation is faithful, then A supports B in the sense of the Cuntz preorder. The almost extension property implies aperiodicity, and the converse holds if B is separable. A crossed product inclusion has the almost extension property if and only if the dual groupoid of the action is topologically principal. Topologically free actions are always aperiodic. If A is separable or of Type I, then topological freeness, aperiodicity and having a unique pseudo-expectation are equivalent to the condition that A detects ideals in all intermediate C*-algebras. If, in addition, B is separable, then all these conditions are equivalent to the almost extension property.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133017041","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 : 2020-06-24DOI: 10.1142/S0219025721500120
Floris Elzinga
Recently, Brannan and Vergnioux showed that the free orthogonal quantum group factors $mathcal{L}mathbb{F}O_M$ have Jung's strong 1-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in 2N dimensions $J_{2N}$. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in 1-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a 1-bounded set without losing 1-boundedness. In particular this allows us to include the character of the fundamental representation, proving strong 1-boundedness.
{"title":"Strong 1-boundedness of unimodular orthogonal free quantum groups","authors":"Floris Elzinga","doi":"10.1142/S0219025721500120","DOIUrl":"https://doi.org/10.1142/S0219025721500120","url":null,"abstract":"Recently, Brannan and Vergnioux showed that the free orthogonal quantum group factors $mathcal{L}mathbb{F}O_M$ have Jung's strong 1-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in 2N dimensions $J_{2N}$. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in 1-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a 1-bounded set without losing 1-boundedness. In particular this allows us to include the character of the fundamental representation, proving strong 1-boundedness.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130212441","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}
A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be $mathbb R$-linear. On the opposite direction of this result, we introduce a notion of embeddability between operator spaces and show that this notion is strictly weaker than complete $mathbb R$-isomorphic embeddability (in particular, weaker than complete $mathbb C$-isomorphic embeddability). Although weaker, this notion is strong enough for some applications. For instance, we show that if an infinite dimensional operator space $X$ embeds in this weaker sense into Pisier's operator space $mathrm{OH}$, then $X$ must be completely isomorphic to $mathrm{OH}$.
{"title":"Completely coarse maps are ${mathbb {R}}$-linear","authors":"B. M. Braga, J. A. Chávez-Domínguez","doi":"10.1090/proc/15289","DOIUrl":"https://doi.org/10.1090/proc/15289","url":null,"abstract":"A map between operator spaces is called completely coarse if the sequence of its amplifications is equi-coarse. We prove that all completely coarse maps must be $mathbb R$-linear. On the opposite direction of this result, we introduce a notion of embeddability between operator spaces and show that this notion is strictly weaker than complete $mathbb R$-isomorphic embeddability (in particular, weaker than complete $mathbb C$-isomorphic embeddability). Although weaker, this notion is strong enough for some applications. For instance, we show that if an infinite dimensional operator space $X$ embeds in this weaker sense into Pisier's operator space $mathrm{OH}$, then $X$ must be completely isomorphic to $mathrm{OH}$.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131381431","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 class of AD algebras of real rank zero is classified by an exact sequence of K-groups with coefficients, equipped with certain order structures. Such a sequence is always split, and one may ask why, then, the middle group is relevant for classification. The answer is that the splitting map can not always be chosen to respect the order structures involved. This may be rephrased in terms of the ideals of the C*-algebras in question. We prove that when the C*-algebra has only finitely many ideals, a splitting map respecting these always exists. Hence AD algebras of real rank zero with finitely many ideals are classified by (classical) ordered K-theory. We also indicate how the methods generalize to the full class of ASH algebras with slow dimension growth and real rank zero.
{"title":"K\"unneth Splittings and Classification of C*-Algebras with Finitely Many Ideals.","authors":"S. Eilers","doi":"10.1090/fic/013/04","DOIUrl":"https://doi.org/10.1090/fic/013/04","url":null,"abstract":"The class of AD algebras of real rank zero is classified by an exact sequence of K-groups with coefficients, equipped with certain order structures. Such a sequence is always split, and one may ask why, then, the middle group is relevant for classification. The answer is that the splitting map can not always be chosen to respect the order structures involved. \u0000This may be rephrased in terms of the ideals of the C*-algebras in question. We prove that when the C*-algebra has only finitely many ideals, a splitting map respecting these always exists. Hence AD algebras of real rank zero with finitely many ideals are classified by (classical) ordered K-theory. We also indicate how the methods generalize to the full class of ASH algebras with slow dimension growth and real rank zero.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122685195","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 : 2020-05-21DOI: 10.1142/S1793525321500382
J. A. Chávez-Domínguez, A. Swift
We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete.
{"title":"Asymptotic dimension and coarse embeddings in the quantum setting","authors":"J. A. Chávez-Domínguez, A. Swift","doi":"10.1142/S1793525321500382","DOIUrl":"https://doi.org/10.1142/S1793525321500382","url":null,"abstract":"We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of reflexive quantum expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124462642","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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