Operator system quotiens of matrix algebras and their tensor products
- Autores: Douglas Farenick, Vern I. Paulsen
- Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 111, Nº 2, 2012, págs. 210-243
- Idioma: inglés
- Enlaces
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Resumen
{If} ϕ:S→T is a completely positive (cp) linear map of operator systems and if J=kerϕ, then the quotient vector space S/J may be endowed with a matricial ordering through which S/J has the structure of an operator system. Furthermore, there is a uniquely determined cp map ϕ˙:S/J→T such that ϕ=ϕ˙∘q, where q is the canonical linear map of S onto S/J. The cp map ϕ is called a complete quotient map if ϕ˙ is a complete order isomorphism between the operator systems S/J and T. Herein we study certain quotient maps in the cases where 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 C∗(F∞)⊗minB(H)=C∗(F∞)⊗maxB(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 C∗-algebras that have the weak expectation property.
