Connes' Embedding Problem and Lance's WEP

On a Characterization of the Weak Expectation Property (WEP)

We give a detailed proof of a new characterization of the Weak Expectation Property (WEP) announced by Haagerup in the 1990's but unavailable (in any form) till now. Our main result is motivated by a well known conjecture of Kirchberg, which is equivalent to the Connes embedding problem. We review the basic relevant facts connecting our main theorem with the latter conjecture, along the lines of our forthcoming lecture notes volume on the Connes-Kirchberg problem.

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

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 …

On Kirchberg's Embedding Problem

Kirchberg's Embedding Problem (KEP) asks whether every separable C$^*$ algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses. In order to prove this result, we study general properties of existentially closed C$^*$ algebras. Along the way, we establish a connection between existentially closed C$^*$ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper c…

关于强相似性及Connes嵌入问题的注记

A note on strong similarity and the Connes embedding problem

本文研究了强相似性与Connes嵌入问题，证明了某些完全有界同态无法被正交化，且不具有特定提升性质。