scholarly journals Strongly Peaking Representations and Compressions of Operator Systems

2020 ◽  
Author(s):  
Kenneth R Davidson ◽  
Benjamin Passer
Keyword(s):  
Convex Sets ◽  
Uniqueness Theorems ◽  
Unitary Equivalence ◽  
Operator System ◽  
Direct Sum ◽  
Operator Systems ◽  
Minimal Presentations ◽  
Minimality Conditions ◽  
Additional Assumptions

Abstract We use Arveson’s notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets that admit minimal presentations. A fully compressed separable operator system necessarily generates the $C^*$-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.

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 Homometry and Direct-Sum Decompositions of Lattice-Convex Sets

2016 ◽  
Vol 56 (1) ◽  
pp. 216-249
Author(s):  
Gennadiy Averkov ◽  
Barbara Langfeld
Keyword(s):  
Convex Sets ◽  
Direct Sum

QUOTIENT AND PSEUDO UNIT IN NONUNITAL OPERATOR SYSTEM

2015 ◽  
Vol 92 (1) ◽  
pp. 123-132
Author(s):  
LI LIU ◽  
JIAN-ZE LI
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
System Structure ◽  
Operator System ◽  
Operator Systems ◽  
Necessary And Sufficient

We define the quotient and complete NUOS-quotient map (NUOS stands for nonunital operator system) in the category of nonunital operator systems. We prove that the greatest reduced tensor product max0 is projective in some sense. Moreover, we define a pseudo unit in a nonunital operator system and give some necessary and sufficient conditions under which a nonunital operator system has an operator system structure.

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 The operator system of Toeplitz matrices

2021 ◽  
Vol 8 (32) ◽  
pp. 999-1023
Author(s):  
Douglas Farenick
Keyword(s):  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Matrix Analysis ◽  
Trigonometric Polynomials ◽  
Complex Matrices ◽  
Content Type ◽  
Operator System ◽  
Order Isomorphism ◽  
Positive Linear Map ◽  
Operator Systems

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n × n n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n n . The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n n\times n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n × n n\times n Toeplitz matrices into the algebra of all n × n n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n × n n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξ n \xi _n generates an extremal ray in the cone of all continuous n × n n\times n Toeplitz-matrix valued functions f f on the unit circle S 1 S^1 whose Fourier coefficients f ^ ( k ) \hat f(k) vanish for | k | ≥ n |k|\geq n . Lastly, it is noted that all positive Toeplitz matrices over nuclear C ∗ ^* -algebras are approximately separable.

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).

EMBEDDINGS AND -ENVELOPES OF EXACT OPERATOR SYSTEMS

2017 ◽  
Vol 96 (2) ◽  
pp. 274-285
Author(s):  
PREETI LUTHRA ◽  
AJAY KUMAR
Keyword(s):  
Tensor Product ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operator System ◽  
Operator Systems ◽  
Necessary And Sufficient

We prove a necessary and sufficient condition for embeddability of an operator system into ${\mathcal{O}}_{2}$. Using Kirchberg’s theorems on a tensor product of ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{\infty }$, we establish results on their operator system counterparts ${\mathcal{S}}_{2}$ and ${\mathcal{S}}_{\infty }$. Applications of the results, including some examples describing $C^{\ast }$-envelopes of operator systems, are also discussed.

scholarly journals The noncommutative Choquet Boundary III

2010 ◽  
Vol 106 (2) ◽  
pp. 196 ◽  
Author(s):  
William Arveson
Keyword(s):  
Hilbert Spaces ◽  
Banach Algebras ◽  
Operator System ◽  
Choquet Boundary ◽  
Finite Dimensional ◽  
Operator Systems ◽  
Boundary Ideal

We classify operator systems $S\subseteq \mathcal{B}(H)$ that act on finite dimensional Hilbert spaces $H$ by making use of the noncommutative Choquet boundary. $S$ is said to be reduced when its boundary ideal is $\{0\}$. In the category of operator systems, that property functions as semisimplicity does in the category of complex Banach algebras. We construct explicit examples of reduced operator systems using sequences of "parameterizing maps" $\Gamma_k: \mathsf{C}^r\to \mathcal{B}(H_k)$, $k=1,\dots, N$. We show that every reduced operator system is isomorphic to one of these, and that two sequences give rise to isomorphic operator systems if and only if they are "unitarily equivalent" parameterizing sequences. Finally, we construct nonreduced operator systems $S$ that have a given boundary ideal $K$ and a given reduced image in $C^*(S)/K$, and show that these constructed examples exhaust the possibilities.

scholarly journals A decomposition of the tensor product of matrices

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 105-124
Author(s):  
Caixing Gu ◽  
Jaehui Park
Keyword(s):  
Tensor Product ◽  
Unitary Equivalence ◽  
Direct Sum ◽  
Product Of Matrices

In this paper we decompose (under unitary equivalence) the tensor product A ? A into a direct sum of irreducible matrices, when A is a 3 x 3 matrix.

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