scholarly journals OPERATOR SYSTEM NUCLEARITY VIA -ENVELOPES

2016 ◽  
Vol 101 (3) ◽  
pp. 356-375 ◽  
Author(s):  
VED PRAKASH GUPTA ◽  
PREETI LUTHRA
Keyword(s):  
Discrete Group ◽  
Tensor Products ◽  
Detailed Comparison ◽  
Finite Graph ◽  
Discrete Groups ◽  
Generating Set ◽  
Operator System ◽  
Operator Systems ◽  
Minimal Generating Set ◽  
Math Phys

We prove that an operator system is (min, ess)-nuclear if its $C^{\ast }$-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by Farenick et al. [‘Operator systems from discrete groups’, Comm. Math. Phys.329(1) (2014), 207–238] is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete).

An approximation property for operator systems

2019 ◽  
Vol 22 (01) ◽  
pp. 1950005
Author(s):  
Wei Wu
Keyword(s):  
Tensor Product ◽  
Convex Sets ◽  
Approximation Property ◽  
Tensor Products ◽  
Order Unit ◽  
Operator System ◽  
Operator Systems ◽  
Natural Sense

Motivated by an observation of Namioka and Phelps on an approximation property of order unit spaces, we introduce the [Formula: see text]-tensor product and the [Formula: see text]-tensor product of two compact matrix convex sets. We define a new approximation property for operator systems, and give a characterization using the [Formula: see text]- and [Formula: see text]-tensor products in the spirit of Grothendieck. Thus, an operator system has the operator system approximation property if and only if it is [Formula: see text]-nuclear in a natural sense.

scholarly journals Operator system quotients of matrix algebras and their tensor products

2012 ◽  
Vol 111 (2) ◽  
pp. 210 ◽  
Author(s):  
Douglas Farenick ◽  
Vern I. Paulsen
Keyword(s):  
Tensor Products ◽  
Operator System ◽  
Linear Map ◽  
Order Isomorphism ◽  
Matrix Algebras ◽  
C Algebras ◽  
Tridiagonal Matrices ◽  
Quotient Maps ◽  
Operator Systems ◽  
Affirmative Solution

{If} $\phi:\mathcal{S}\rightarrow\mathcal{T}$ is a completely positive (cp) linear map of operator systems and if $\mathcal{J}=\ker\phi$, then the quotient vector space $\mathcal{S}/\mathcal{J}$ may be endowed with a matricial ordering through which $\mathcal{S}/\mathcal{J}$ has the structure of an operator system. Furthermore, there is a uniquely determined cp map $\dot{\phi}:\mathcal{S}/\mathcal{J} \rightarrow\mathcal{T}$ such that $\phi=\dot{\phi}\circ q$, where $q$ is the canonical linear map of $\mathcal{S}$ onto $\mathcal{S}/\mathcal{J}$. The cp map $\phi$ is called a complete quotient map if $\dot{\phi}$ is a complete order isomorphism between the operator systems $\mathcal{S}/\mathcal{J}$ and $\mathcal{T}$. Herein we study certain quotient maps in the cases where $\mathcal{S}$ is a full matrix algebra or a full subsystem of tridiagonal matrices. Our study of operator system quotients of matrix algebras and tensor products has applications to operator algebra theory. In particular, we give a new, simple proof of Kirchberg's Theorem $\operatorname{C}^*(\mathbf{F}_\infty)\otimes_{\min}\mathcal{B}(\mathcal{H})=\operatorname{C}^*(\mathbf{F}_\infty)\otimes_{\max}\mathcal{B}(\mathcal{H})$, show that an affirmative solution to the Connes Embedding Problem is implied by various matrix-theoretic problems, and give a new characterisation of unital $\operatorname{C}^*$-algebras that have the weak expectation property.

scholarly journals Clebsch–Gordan coefficients of discrete groups in subgroup bases

2018 ◽  
Vol 33 (10) ◽  
pp. 1850055
Author(s):  
Gaoli Chen
Keyword(s):  
Discrete Group ◽  
Irreducible Representations ◽  
Complex Conjugate ◽  
Discrete Groups

We express each Clebsch–Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to sign ambiguities. We describe the procedure of obtaining embedding factors and then calculate CG coefficients of the group [Formula: see text] in terms of embedding factors of its subgroups [Formula: see text] and [Formula: see text].

scholarly journals The Fundamental Group of Balanced Simplicial Complexes and Posets

10.37236/73 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Steven Klee
Keyword(s):  
Fundamental Group ◽  
Upper Bound ◽  
Large Family ◽  
Simplicial Complexes ◽  
Generating Set ◽  
Minimal Generating Set

We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

scholarly journals On the density of Cayley graphs of R.Thompson’s group F in symmetric generators

2021 ◽  
pp. 1-13
Author(s):  
V. S. Guba
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Finite Graph ◽  
Vertex Degree ◽  
Doubling Property ◽  
Generating Set ◽  
Finite Set ◽  
Standard Set ◽  
Thompson’S Group ◽  
Average Vertex

By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property.

The Fuzziness of a Minimal Generating Set of Fedorov Groups

2021 ◽  
Vol 66 (6) ◽  
pp. 913-919
Author(s):  
A. M. Banaru ◽  
V. R. Shiroky ◽  
D. A. Banaru
Keyword(s):  
Generating Set ◽  
Minimal Generating Set

The Number of Generators of a Linear p-Group

1972 ◽  
Vol 24 (5) ◽  
pp. 851-858 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Generating Set ◽  
Nilpotence Class ◽  
Number Of Generators ◽  
Minimal Generating Set

Let G be a finite p-group, having a faithful character χ of degree f. The object of this paper is to bound the number, d(G), of generators in a minimal generating set for G in terms of χ and in particular in terms of f. This problem was raised by D. M. Goldschmidt, and solved by him in the case that G has nilpotence class 2.

scholarly journals Certain unitary representations of the infinite symmetric group, II

1987 ◽  
Vol 106 ◽  
pp. 143-162 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Type I ◽  
Infinite Symmetric Group ◽  
Locally Finite ◽  
Distinctive Features ◽  
Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

scholarly journals Quasi-flat representations of uniform groups and quantum groups

2019 ◽  
Vol 18 (08) ◽  
pp. 1950155 ◽  
Author(s):  
Teodor Banica ◽  
Alexandru Chirvasitu
Keyword(s):  
Quantum Groups ◽  
Discrete Group ◽  
Homogeneous Spaces ◽  
Model Space ◽  
Representation Formula ◽  
Discrete Groups ◽  
Universal Model ◽  
Induced Representation ◽  
Negative Results ◽  
Number Formula

Given a discrete group [Formula: see text] and a number [Formula: see text], a unitary representation [Formula: see text] is called quasi-flat when the eigenvalues of each [Formula: see text] are uniformly distributed among the [Formula: see text]th roots of unity. The quasi-flat representations of [Formula: see text] form altogether a parametric matrix model [Formula: see text]. We compute here the universal model space [Formula: see text] for various classes of discrete groups, notably with results in the case where [Formula: see text] is metabelian. We are particularly interested in the case where [Formula: see text] is a union of compact homogeneous spaces, and where the induced representation [Formula: see text] is stationary in the sense that it commutes with the Haar functionals. We present several positive and negative results on this subject. We also discuss similar questions for the discrete quantum groups, proving a stationarity result for the discrete dual of the twisted orthogonal group [Formula: see text].

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