scholarly journals On maximal tensor products and quotient maps of operator systems

2011 ◽  
Vol 384 (2) ◽  
pp. 375-386 ◽  
Author(s):  
Kyung Hoon Han
Keyword(s):  
Tensor Products ◽  
Quotient Maps ◽  
Operator Systems

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.

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.

Tensor products of hyperrigid operator systems

2018 ◽  
Vol 9 (3) ◽  
pp. 369-375 ◽  
Author(s):  
P. Shankar ◽  
A. K. Vijayarajan
Keyword(s):  
Tensor Products ◽  
Operator Systems

scholarly journals Tensor products for non-unital operator systems

2012 ◽  
Vol 396 (2) ◽  
pp. 601-605 ◽  
Author(s):  
Jian-Ze Li ◽  
Chi-Keung Ng
Keyword(s):  
Tensor Products ◽  
Operator Systems

scholarly journals Tensor products of operator systems

2011 ◽  
Vol 261 (2) ◽  
pp. 267-299 ◽  
Author(s):  
Ali Kavruk ◽  
Vern I. Paulsen ◽  
Ivan G. Todorov ◽  
Mark Tomforde
Keyword(s):  
Tensor Products ◽  
Operator Systems

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).

scholarly journals Tropical ideals do not realise all Bergman fans

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Jan Draisma ◽  
Felipe Rincón
Keyword(s):  
Tropical Geometry ◽  
Tensor Products ◽  
Tropical Variety ◽  
Polyhedral Complex ◽  
Direct Sum ◽  
Weight One ◽  
Polyhedral Complexes

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

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