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

By Douglas R. Farenick, A. Kavruk, Vern I. Paulsen. 2011

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A classic theorem of T. Ando characterises operators that have numerical radius at most one as operators that admit a certain positive 2x2 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 nxn 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 A of B(H) has WEP if and only if whenever a certain 3x3 (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 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 3x3 matrix completions.

Subfields: Algebra and Number Theory, Materials Chemistry, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/725c7993e38f62fac6d739388b2232dbe2345d94

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In Algebra and Number Theory

In Materials Chemistry

A. Kavruk

In Mathematical Physics

Citations in the corpus

Characterisations of the weak expectation property (2013, result): “These results expand on earlier work of the first three authors [8] and of the second author [15] that also obtained such representation-free characterisations of WEP.”
On Kirchberg's Embedding Problem (2014, background): “is an easy exercise to check that these two notions of strictly positive agree for elements of unital C∗ algebras. We need the following recent theorem of Farenick, Kavruk, and Paulsen. Theorem 2.25 ([15]). Suppose that A⊆ B(H) is a unital C∗ algebra. The following are equivalent: (1) Ahas WEP; 14 I”
Nuclearity Related Properties in Operator Systems (2011, background): “ on S⊗S 2. Since min ≤c we obtain that Shas property S 2. The proof for Tis similar. The fact that T 3/J 3 = S 2 together with Theorem 5.9 allow us characterize WEP as follows (we refer the reader to [11] for further applications of this result): Theorem 10.10. A unital C*-algebra Ahas WEP if and on”