Journal of Operator Theory

Volume 70, Issue 2, Autumn 2013 pp. 573-590.

$C^*$ -algebras with the weak expectation property and a multivariable analogue of Ando's theorem on the numerical radius

Authors: Douglas Farenick (1), Ali S. Kavruk (2), and Vern I. Paulsen (3)
Author institution: (1) Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada
(2) Department of Mathematics, University of Houston, Houston, Texas 77204-3476, U.S.A.
(3) Department of Mathematics, University of Houston, Houston, Texas 77204-3476, U.S.A.

Summary: A classic theorem of T.~Ando characterises operators that have numerical radius at most one as operators that admit a certain positive

$2\times 2$ operator matrix completion. In this paper we consider variants of Ando's theorem in which the operators (and matrix completions) are constrained to a given

$C^*$ -algebra. By considering

$n\times n$ matrix completions, an extension of Ando's theorem to a multivariable setting is made. We show that the

$C^*$ -algebras in which these extended formulations of Ando's theorem hold true are precisely the

$C^*$ -algebras with the weak expectation property (WEP). We also show that a

$C^*$ -subalgebra of

$\bh$ has WEP if and only if whenever a certain

$3\times 3$ (operator) matrix completion problem can be solved in matrices over

$\bh$ , it can also be solved in matrices over

$\csta$ . This last result gives a characterisation of WEP that is spatial and yet is independent of the particular representation of the

$C^*$ -algebra. This leads to a new characterisation of injective von Neumann algebras. We also give a new equivalent formulation of the Connes embedding problem as a problem concerning

$3\times3$ matrix completions.

DOI: http://dx.doi.org/10.7900/jot.2011oct07.1938
Keywords: weak expectation property, Ando's theorem, the Connes embedding problem, numerical radius, operator system quotient, operator system tensor product

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