We develop a completely bounded counterpart to the non-commutative Choquet boundary of an operator space. We show how the class of completely bounded linear maps is too large to accommodate our purposes. To overcome this obstacle, we isolate the subset of completely bounded linear maps on an operator space admitting a dilation of the same norm which is multiplicative on the generated $C^*$-algebra. We view such maps as analogues of the familiar unital completely contractive maps, and we exhibit many of their structural properties. Of particular interest to us are those maps which are extremal with respect to a natural dilation order. We establish the existence of extremals and show that they have a certain unique extension property. In particular, they give rise to $*$-homomorphisms which we use to associate to any representation of an operator space an entire scale of $C^*$-envelopes. We conjecture that these $C^*$-envelopes are all $*$-isomorphic, and verify this in some important cases.
{"title":"A completely bounded non-commutative Choquet boundary for operator spaces","authors":"Raphael Clouatre, Christopher Ramsey","doi":"10.1093/IMRN/RNX335","DOIUrl":"https://doi.org/10.1093/IMRN/RNX335","url":null,"abstract":"We develop a completely bounded counterpart to the non-commutative Choquet boundary of an operator space. We show how the class of completely bounded linear maps is too large to accommodate our purposes. To overcome this obstacle, we isolate the subset of completely bounded linear maps on an operator space admitting a dilation of the same norm which is multiplicative on the generated $C^*$-algebra. We view such maps as analogues of the familiar unital completely contractive maps, and we exhibit many of their structural properties. Of particular interest to us are those maps which are extremal with respect to a natural dilation order. We establish the existence of extremals and show that they have a certain unique extension property. In particular, they give rise to $*$-homomorphisms which we use to associate to any representation of an operator space an entire scale of $C^*$-envelopes. We conjecture that these $C^*$-envelopes are all $*$-isomorphic, and verify this in some important cases.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125849192","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}
It is proved that for every surjective linear isometry $V$ on a perfect Banach symmetric ideal $mathcal C_Eneq mathcal C_2$ of compact operators, acting in a complex separable infnite-dimensional Hilbert space $mathcal H$ there exist unitary operators $u$ and $v$ on $mathcal H$ such that $V(x)=uxv$ or $V(x) = ux^tv$ for all $xin mathcal C_E$, where $x^t$ is the transpose of an operator $x$ with respect to a fixed orthonormal basis in $mathcal H$. In addition, it is shown that any surjective 2-local isometry on a perfect Banach symmetric ideal $mathcal C_E neq mathcal C_2$ is a linear isometry on $mathcal C_E$.
{"title":"Isometries of perfect norm ideals of compact operators","authors":"B. Aminov, V. Chilin","doi":"10.4064/SM170306-19-4","DOIUrl":"https://doi.org/10.4064/SM170306-19-4","url":null,"abstract":"It is proved that for every surjective linear isometry $V$ on a perfect Banach symmetric ideal $mathcal C_Eneq mathcal C_2$ of compact operators, acting in a complex separable infnite-dimensional Hilbert space $mathcal H$ there exist unitary operators $u$ and $v$ on $mathcal H$ such that $V(x)=uxv$ or $V(x) = ux^tv$ for all $xin mathcal C_E$, where $x^t$ is the transpose of an operator $x$ with respect to a fixed orthonormal basis in $mathcal H$. In addition, it is shown that any surjective 2-local isometry on a perfect Banach symmetric ideal $mathcal C_E neq mathcal C_2$ is a linear isometry on $mathcal C_E$.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129706277","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}
Random matrices like GUE, GOE and GSE have been studied for decades and have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and Thorbj{o}rnsen in their paper [18], it is called strong convergence property and then more random matrices with this property are followed (see [27], [5], [1], [24], [10] and [3]). In general, the definition can be stated for a sequence of tuples over some text{C}^{ast}-algebras. And in this general setting, some stability property under reduced free product can be achieved (see Skoufranis [30] and Pisier [26]), as an analogy of the result by Camille Male [24] for random matrices. In this paper, we want to show that, for a sequence of strongly convergent random variables, non-commutative polynomials can be extended to non-commutative rational functions under certain assumptions. Roughly speaking, the strong convergence property is stable under taking the inverse. As a direct corollary, we can conclude that for a tuple (X_{1}^{left(nright)},cdots,X_{m}^{left(nright)}) of independent GUE random matrices, r(X_{1}^{left(nright)},cdots,X_{m}^{left(nright)}) converges in trace and in norm to r(s_{1},cdots,s_{m}) almost surely, where r is a rational function and (s_{1},cdots,s_{m}) is a tuple of freely independent semi-circular elements which lies in the domain of r.
{"title":"Non-commutative rational function in strongly convergent random variables","authors":"Sheng Yin","doi":"10.22034/aot.1702-1126","DOIUrl":"https://doi.org/10.22034/aot.1702-1126","url":null,"abstract":"Random matrices like GUE, GOE and GSE have been studied for decades and have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and Thorbj{o}rnsen in their paper [18], it is called strong convergence property and then more random matrices with this property are followed (see [27], [5], [1], [24], [10] and [3]). In general, the definition can be stated for a sequence of tuples over some text{C}^{ast}-algebras. And in this general setting, some stability property under reduced free product can be achieved (see Skoufranis [30] and Pisier [26]), as an analogy of the result by Camille Male [24] for random matrices. \u0000In this paper, we want to show that, for a sequence of strongly convergent random variables, non-commutative polynomials can be extended to non-commutative rational functions under certain assumptions. Roughly speaking, the strong convergence property is stable under taking the inverse. As a direct corollary, we can conclude that for a tuple (X_{1}^{left(nright)},cdots,X_{m}^{left(nright)}) of independent GUE random matrices, r(X_{1}^{left(nright)},cdots,X_{m}^{left(nright)}) converges in trace and in norm to r(s_{1},cdots,s_{m}) almost surely, where r is a rational function and (s_{1},cdots,s_{m}) is a tuple of freely independent semi-circular elements which lies in the domain of r.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126776110","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 : 2017-02-07DOI: 10.2140/pjm.2018.292.203
S. Palcoux
Ore proved that a finite group is cyclic if and only if its subgroup lattice is distributive. Now, since every subgroup of a cyclic group is normal, we call a subfactor planar algebra cyclic if all its biprojections are normal and form a distributive lattice. The main result generalizes one side of Ore's theorem and shows that a cyclic subfactor is singly generated in the sense that there is a minimal 2-box projection generating the identity biprojection. We conjecture that this result holds without assuming the biprojections to be normal, and we show that it is true for small lattices. We finally exhibit a dual version of another theorem of Ore and a non-trivial upper bound for the minimal number of irreducible components for a faithful complex representation of a finite group.
{"title":"Ore's theorem on cyclic subfactor planar algebras and beyond","authors":"S. Palcoux","doi":"10.2140/pjm.2018.292.203","DOIUrl":"https://doi.org/10.2140/pjm.2018.292.203","url":null,"abstract":"Ore proved that a finite group is cyclic if and only if its subgroup lattice is distributive. Now, since every subgroup of a cyclic group is normal, we call a subfactor planar algebra cyclic if all its biprojections are normal and form a distributive lattice. The main result generalizes one side of Ore's theorem and shows that a cyclic subfactor is singly generated in the sense that there is a minimal 2-box projection generating the identity biprojection. We conjecture that this result holds without assuming the biprojections to be normal, and we show that it is true for small lattices. We finally exhibit a dual version of another theorem of Ore and a non-trivial upper bound for the minimal number of irreducible components for a faithful complex representation of a finite group.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116620356","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 quantitative Fourth Moment Theorem for Wigner integrals of any order with symmetric kernels, generalizing an earlier result from Kemp et al. (2012). The proof relies on free stochastic analysis and uses a new biproduct formula for bi-integrals. A consequence of our main result is a Nualart-Ortiz-Latorre type characterization of convergence in law to the semicircular distribution for Wigner integrals. As an application, we provide Berry-Esseen type bounds in the context of the free Breuer-Major theorem for the free fractional Brownian motion.
我们证明了具有对称核的任意阶Wigner积分的定量第四矩定理，推广了Kemp et al.(2012)的早期结果。该证明依赖于自由随机分析，并使用了双积分的一个新的双积公式。我们的主要结果的一个结果是对Wigner积分的半圆分布收敛规律的nualart - ortize - latorre型表征。作为一个应用，我们给出了自由分数阶布朗运动的自由Breuer-Major定理背景下的Berry-Esseen型界。
{"title":"Free quantitative fourth moment theorems on Wigner space","authors":"S. Bourguin, Simon Campese","doi":"10.1093/IMRN/RNX036","DOIUrl":"https://doi.org/10.1093/IMRN/RNX036","url":null,"abstract":"We prove a quantitative Fourth Moment Theorem for Wigner integrals of any order with symmetric kernels, generalizing an earlier result from Kemp et al. (2012). The proof relies on free stochastic analysis and uses a new biproduct formula for bi-integrals. A consequence of our main result is a Nualart-Ortiz-Latorre type characterization of convergence in law to the semicircular distribution for Wigner integrals. As an application, we provide Berry-Esseen type bounds in the context of the free Breuer-Major theorem for the free fractional Brownian motion.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117353266","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 : 2017-01-16DOI: 10.1216/RMJ-2017-47-8-2699
W. Lewkeeratiyutkul, Saeid Zahmatkesh
Let $(A,alpha)$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $alpha$ of $A$. We describe the primitive ideal space of the partial-isometric crossed product $Atimes_{alpha}^{textrm{piso}}mathbb{N}$ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.
{"title":"The primitive ideal space of the partial-isometric crossed product of a system by a single automorphism","authors":"W. Lewkeeratiyutkul, Saeid Zahmatkesh","doi":"10.1216/RMJ-2017-47-8-2699","DOIUrl":"https://doi.org/10.1216/RMJ-2017-47-8-2699","url":null,"abstract":"Let $(A,alpha)$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $alpha$ of $A$. We describe the primitive ideal space of the partial-isometric crossed product $Atimes_{alpha}^{textrm{piso}}mathbb{N}$ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132395144","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}
Trace formulas are investigated in non-commutative integration theory. The main result is to evaluate the standard trace of a Takesaki dual and, for this, we introduce the notion of interpolator and accompanied boundary objects. The formula is then applied to explore a variation of Haagerup's trace formula.
{"title":"Around trace formulas in non-commutative integration","authors":"S. Yamagami","doi":"10.4171/PRIMS/54-1-7","DOIUrl":"https://doi.org/10.4171/PRIMS/54-1-7","url":null,"abstract":"Trace formulas are investigated in non-commutative integration theory. The main result is to evaluate the standard trace of a Takesaki dual and, for this, we introduce the notion of interpolator and accompanied boundary objects. The formula is then applied to explore a variation of Haagerup's trace formula.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124492070","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 : 2017-01-08DOI: 10.2140/pjm.2018.293.301
S. Kaliszewski, M. B. Landstad, John Quigg
In further study of the application of crossed-product functors to the Baum-Connes Conjecture, Buss, Echterhoff, and Willett introduced various other properties that crossed-product functors may have. Here we introduce and study analogues of these properties for coaction functors, making sure that the properties are preserved when the coaction functors are composed with the full crossed product to make a crossed-product functor. The new properties for coaction functors studied here are functoriality for generalized homomorphisms and the correspondence property. We particularly study the connections with the ideal property. The study of functoriality for generalized homomorphisms requires a detailed development of the Fischer construction of maximalization of coactions with regard to possibly degenerate homomorphisms into multiplier algebras. We verify that all "KLQ" functors arising from large ideals of the Fourier-Stieltjes algebra $B(G)$ have all the properties we study, and at the opposite extreme we give an example of a coaction functor having none of the properties.
{"title":"Coaction functors, II","authors":"S. Kaliszewski, M. B. Landstad, John Quigg","doi":"10.2140/pjm.2018.293.301","DOIUrl":"https://doi.org/10.2140/pjm.2018.293.301","url":null,"abstract":"In further study of the application of crossed-product functors to the Baum-Connes Conjecture, Buss, Echterhoff, and Willett introduced various other properties that crossed-product functors may have. Here we introduce and study analogues of these properties for coaction functors, making sure that the properties are preserved when the coaction functors are composed with the full crossed product to make a crossed-product functor. The new properties for coaction functors studied here are functoriality for generalized homomorphisms and the correspondence property. We particularly study the connections with the ideal property. The study of functoriality for generalized homomorphisms requires a detailed development of the Fischer construction of maximalization of coactions with regard to possibly degenerate homomorphisms into multiplier algebras. We verify that all \"KLQ\" functors arising from large ideals of the Fourier-Stieltjes algebra $B(G)$ have all the properties we study, and at the opposite extreme we give an example of a coaction functor having none of the properties.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131270756","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 an algorithm to compute the $K$-groups of the crossed product by the flip automorphism for a nuclear C$^*$-algebra satisfying the UCT.
对于满足UCT的核C$^*$-代数，给出了用翻转自同构计算叉积的$K$-群的算法。
{"title":"The K-theory of the flip automorphisms","authors":"Masaki Izumi","doi":"10.2969/aspm/08010123","DOIUrl":"https://doi.org/10.2969/aspm/08010123","url":null,"abstract":"We give an algorithm to compute the $K$-groups of the crossed product by the flip automorphism for a nuclear C$^*$-algebra satisfying the UCT.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132596815","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 : 2017-01-05DOI: 10.22130/SCMA.2018.73698.306
K. Nourouzi, A. Reza
In this paper we give three functors $mathfrak{P}$, $[cdot]_K$ and $mathfrak{F}$ on the category of C$^ast$-algebras. The functor $mathfrak{P}$ assigns to each C$^ast$-algebra $mathcal{A}$ a pre-C$^ast$-algebra $mathfrak{P}(mathcal{A})$ with completion $[mathcal{A}]_K$. The functor $[cdot]_K$ assigns to each C$^ast$-algebra $mathcal{A}$ the Cauchy extension $[mathcal{A}]_K$ of $mathcal{A}$ by a non-unital C$^ast$-algebra $mathfrak{F}(mathcal{A})$. Some properties of these functors are also given. In particular, we show that the functors $[cdot]_K$ and $mathfrak{F}$ are exact and the functor $mathfrak{P}$ is normal exact.
{"title":"Functors induced by Cauchy extension of C*-algebras","authors":"K. Nourouzi, A. Reza","doi":"10.22130/SCMA.2018.73698.306","DOIUrl":"https://doi.org/10.22130/SCMA.2018.73698.306","url":null,"abstract":"In this paper we give three functors $mathfrak{P}$, $[cdot]_K$ and $mathfrak{F}$ on the category of C$^ast$-algebras. The functor $mathfrak{P}$ assigns to each C$^ast$-algebra $mathcal{A}$ a pre-C$^ast$-algebra $mathfrak{P}(mathcal{A})$ with completion $[mathcal{A}]_K$. The functor $[cdot]_K$ assigns to each C$^ast$-algebra $mathcal{A}$ the Cauchy extension $[mathcal{A}]_K$ of $mathcal{A}$ by a non-unital C$^ast$-algebra $mathfrak{F}(mathcal{A})$. Some properties of these functors are also given. In particular, we show that the functors $[cdot]_K$ and $mathfrak{F}$ are exact and the functor $mathfrak{P}$ is normal exact.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122480278","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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