We present a systematic development of inductive limits in the categories of ordered *-vector spaces, Archimedean order unit spaces, matrix ordered spaces, operator systems and operator C*-systems. We show that the inductive limit intertwines the operation of passing to the maximal operator system structure of an Archimedean order unit space, and that the same holds true for the minimal operator system structure if the connecting maps are complete order embeddings. We prove that the inductive limit commutes with the operation of taking the maximal tensor product with another operator system, and establish analogous results for injective functorial tensor products provided the connecting maps are complete order embeddings. We identify the inductive limit of quotient operator systems as a quotient of the inductive limit, in case the involved kernels are completely biproximinal. We describe the inductive limit of graph operator systems as operator systems of topological graphs, show that two such operator systems are completely order isomorphic if and only if their underlying graphs are isomorphic, identify the C*-envelope of such an operator system, and prove a version of Glimm's Theorem on the isomorphism of UHF algebras in the category of operator systems.
{"title":"Inductive limits in the operator system and related categories","authors":"Linda Mawhinney, I. Todorov","doi":"10.4064/DM771-4-2018","DOIUrl":"https://doi.org/10.4064/DM771-4-2018","url":null,"abstract":"We present a systematic development of inductive limits in the categories of ordered *-vector spaces, Archimedean order unit spaces, matrix ordered spaces, operator systems and operator C*-systems. We show that the inductive limit intertwines the operation of passing to the maximal operator system structure of an Archimedean order unit space, and that the same holds true for the minimal operator system structure if the connecting maps are complete order embeddings. We prove that the inductive limit commutes with the operation of taking the maximal tensor product with another operator system, and establish analogous results for injective functorial tensor products provided the connecting maps are complete order embeddings. We identify the inductive limit of quotient operator systems as a quotient of the inductive limit, in case the involved kernels are completely biproximinal. We describe the inductive limit of graph operator systems as operator systems of topological graphs, show that two such operator systems are completely order isomorphic if and only if their underlying graphs are isomorphic, identify the C*-envelope of such an operator system, and prove a version of Glimm's Theorem on the isomorphism of UHF algebras in the category of operator systems.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126608128","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 present a combinatorial approach to the opposite 2-variable bi-free partial $S$-transforms where the opposite multiplication is used on the right. In addition, extensions of this partial $S$-transforms to the conditional bi-free and operator-valued bi-free settings are discussed.
{"title":"A Combinatorial Approach to the Opposite Bi-Free Partial $S$-Transform","authors":"P. Skoufranis","doi":"10.7153/oam-2018-12-22","DOIUrl":"https://doi.org/10.7153/oam-2018-12-22","url":null,"abstract":"In this paper, we present a combinatorial approach to the opposite 2-variable bi-free partial $S$-transforms where the opposite multiplication is used on the right. In addition, extensions of this partial $S$-transforms to the conditional bi-free and operator-valued bi-free settings are discussed.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133513745","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 a relationship between the ultraproduct of a crossed product von Neumann algebra and the crossed product of an ultraproduct von Neumann algebra. As an application, the continuous core of an ultraproduct von Neumann algebra is described.
{"title":"Ultraproducts of crossed product von Neumann algebras","authors":"Reiji Tomatsu","doi":"10.1215/IJM/1534924828","DOIUrl":"https://doi.org/10.1215/IJM/1534924828","url":null,"abstract":"We study a relationship between the ultraproduct of a crossed product von Neumann algebra and the crossed product of an ultraproduct von Neumann algebra. As an application, the continuous core of an ultraproduct von Neumann algebra is described.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"95 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129562191","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 that any type ${rm III_1}$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $ast$-subalgebra that is the range of a normal conditional expectation. We also investigate stability properties of CBP under finite index extensions/restrictions of type ${rm III_1}$ factors.
{"title":"Singular MASAs in type III factors and Connes' Bicentralizer Property","authors":"Cyril Houdayer, S. Popa","doi":"10.2969/ASPM/08010109","DOIUrl":"https://doi.org/10.2969/ASPM/08010109","url":null,"abstract":"We show that any type ${rm III_1}$ factor with separable predual satisfying Connes' Bicentralizer Property (CBP) has a singular maximal abelian $ast$-subalgebra that is the range of a normal conditional expectation. We also investigate stability properties of CBP under finite index extensions/restrictions of type ${rm III_1}$ factors.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117044800","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 $mathcal{G}$ be an ultragraph and let $C^*(mathcal{G})$ be the associated $C^*$-algebra introduced by Mark Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*(mathcal{G})$, we approach the quotient $C^*$-algebra $C^*(mathcal{G})/I_{(H,B)}$ by the $C^*$-algebra of finite graphs and prove versions of gauge invariant and Cuntz-Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph $C^*$-algebras (in the sense of Kirchberg-R${o}$rdam) via Fell bundles.
{"title":"Primitive ideals and pure infiniteness of ultragraph $C^*$-algebras","authors":"H. Larki","doi":"10.4134/JKMS.J170579","DOIUrl":"https://doi.org/10.4134/JKMS.J170579","url":null,"abstract":"Let $mathcal{G}$ be an ultragraph and let $C^*(mathcal{G})$ be the associated $C^*$-algebra introduced by Mark Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*(mathcal{G})$, we approach the quotient $C^*$-algebra $C^*(mathcal{G})/I_{(H,B)}$ by the $C^*$-algebra of finite graphs and prove versions of gauge invariant and Cuntz-Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph $C^*$-algebras (in the sense of Kirchberg-R${o}$rdam) via Fell bundles.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131550414","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 define a relation < for dual operator algebras. We say that B < A if there exists a projection p in A such that B and pAp are Morita equivalent in our sense. We show that < is transitive, and we investigate the following question: If A < B and B < A, then is it true that A and B are stably isomorphic? We propose an analogous relation < for dual operator spaces, and we present some properties of < in this case.
{"title":"Morita embeddings for dual operator algebras and dual operator spaces","authors":"G. Eleftherakis","doi":"10.4064/SM170427-15-8","DOIUrl":"https://doi.org/10.4064/SM170427-15-8","url":null,"abstract":"We define a relation < for dual operator algebras. We say that B < A if there exists a projection p in A such that B and pAp are Morita equivalent in our sense. We show that < is transitive, and we investigate the following question: If A < B and B < A, then is it true that A and B are stably isomorphic? We propose an analogous relation < for dual operator spaces, and we present some properties of < in this case.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117293693","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}
This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators $E(mathcal{M},tau)$, where $mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $tau$, and $E$ is a symmetric function space. If $Esubset c_0$ is a symmetric sequence space then the analogous properties in the unitary matrix ideals $C_E$ are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Polya, Kothe duality, the spaces $L_p(mathcal{M},tau)$, $1le p
{"title":"Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals","authors":"M. Czerwińska, A. Kamińska","doi":"10.14708/CM.V57I1.3291","DOIUrl":"https://doi.org/10.14708/CM.V57I1.3291","url":null,"abstract":"This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators $E(mathcal{M},tau)$, where $mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $tau$, and $E$ is a symmetric function space. If $Esubset c_0$ is a symmetric sequence space then the analogous properties in the unitary matrix ideals $C_E$ are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Polya, Kothe duality, the spaces $L_p(mathcal{M},tau)$, $1le p<infty$, the identification between $C_E$ and $G(B(H), rm{tr})$ for some symmetric function space $G$, the commutative case when $E$ is identified with $E(mathcal{N}, tau)$ for $mathcal{N}$ isometric to $L_infty$ with the standard integral trace, trace preserving $*$-isomorphisms between $E$ and a $*$-subalgebra of $E(mathcal{M},tau)$, and a general method of removing the assumption of non-atomicity of $mathcal{M}$. The main results on geometric properties are given in separate sections. We present the results on (complex) extreme points, (complex) strict convexity, strong extreme points and midpoint local uniform convexity, $k$-extreme points and $k$-convexity, (complex or local) uniform convexity, smoothness and strong smoothness, (strongly) exposed points, (uniform) Kadec-Klee properties, Banach-Saks properties, Radon-Nikodým property and stability in the sense of Krivine-Maurey. We also state some open problems.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116166656","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 : 2017-03-23DOI: 10.1007/978-3-319-72449-2_4
C. Carvalho, V. Nistor, Yu Qiao
{"title":"Fredholm conditions on non-compact manifolds: theory and examples","authors":"C. Carvalho, V. Nistor, Yu Qiao","doi":"10.1007/978-3-319-72449-2_4","DOIUrl":"https://doi.org/10.1007/978-3-319-72449-2_4","url":null,"abstract":"","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117314848","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 define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products.
{"title":"Infinite alphabet edge shift spaces via ultragraphs and their C*-algebras","authors":"D. Gonccalves, D. Royer","doi":"10.1093/IMRN/RNX175","DOIUrl":"https://doi.org/10.1093/IMRN/RNX175","url":null,"abstract":"We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact. We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132592794","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 $f: mathbb{R}^d tomathbb{R}$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if ${A_k}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]in L_1(H),$ then $$|[f(A_1,cdots,A_d),B]|_{1,infty}leq c(d)|nabla(f)|_{infty}max_{1leq kleq d}|[A_k,B]|_1,$$ where $c(d)$ is a constant independent of $f$, $mathcal{M}$ and $A,B$ and $|cdot|_{1,infty}$ denotes the weak $L_1$-norm. If ${X_k}_{k=1}^d$ (respectively, ${Y_k}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_kin L_1(H),$ then $$|f(X_1,cdots,X_d)-f(Y_1,cdots,Y_d)|_{1,infty}leq c(d)|nabla(f)|_{infty}max_{1leq kleq d}|X_k-Y_k|_1.$$
{"title":"Weak type operator Lipschitz and commutator estimates for commuting tuples","authors":"M. Caspers, F. Sukochev, D. Zanin","doi":"10.5802/AIF.3195","DOIUrl":"https://doi.org/10.5802/AIF.3195","url":null,"abstract":"Let $f: mathbb{R}^d tomathbb{R}$ be a Lipschitz function. If $B$ is a bounded self-adjoint operator and if ${A_k}_{k=1}^d$ are commuting bounded self-adjoint operators such that $[A_k,B]in L_1(H),$ then $$|[f(A_1,cdots,A_d),B]|_{1,infty}leq c(d)|nabla(f)|_{infty}max_{1leq kleq d}|[A_k,B]|_1,$$ where $c(d)$ is a constant independent of $f$, $mathcal{M}$ and $A,B$ and $|cdot|_{1,infty}$ denotes the weak $L_1$-norm. If ${X_k}_{k=1}^d$ (respectively, ${Y_k}_{k=1}^d$) are commuting bounded self-adjoint operators such that $X_k-Y_kin L_1(H),$ then $$|f(X_1,cdots,X_d)-f(Y_1,cdots,Y_d)|_{1,infty}leq c(d)|nabla(f)|_{infty}max_{1leq kleq d}|X_k-Y_k|_1.$$","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130877495","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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