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

Abstract

A classic theorem of T. Ando characterises operators that have numerical radius at most one as operators that admit a certain positive 2 × 2 operator matrix completion. In this paper we consider variants of Ando's the- orem, in which the operators (and matrix completions) are constrained to a given C � -algebra. By considering n × 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 B(H) has WEP if and only there if whenever a certain 3×3 (operator) matrix completion problem can be solved in matrices over B(H), it can also be solved in matrices over A. This last result gives a characterization of WEP that is spatial and yet is independent of the partic- ular representation of the C�-algebra. This leads to a new characterisation of injective von Neumann algebras. We also give a new equivalent formula- tion of the Connes Embedding Problem as a problem concerning 3×3 matrix completions.