# 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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