A Non-Commutative Unitary Analogue of Kirchberg's Conjecture

Abstract

The $C^{\ast }$-algebra ${\mathcal{U}}_{nc}(n)$ is the universal $C^{\ast }$-algebra generated by $n^2$ generators $u_{ij}$ that make up a unitary matrix. We prove that Kirchberg's formulation of Connes' embedding problem has a positive answer if and only if ${\mathcal{U}}_{nc}(2){\otimes }_{\min }{\mathcal{U}}_{nc}(2)={\mathcal{U}}_{nc}(2){\otimes }_{\max }{\mathcal{U}}_{nc}(2)$. Our results follow from properties of the finite-dimensional operator system ${\mathcal{V}}_n$ spanned by $1$ and the generators of ${\mathcal{U}}_{nc}(n)$. We show that ${\mathcal{V}}_n$ is an operator system quotient of $M_{2n}$ and has the OSLLP. We obtain necessary and sufficient conditions on ${\mathcal{V}}_n$ for there to be a positive answer to Kirchberg's problem. Finally, in analogy with recent results of Ozawa, we show that a form of Tsirelson's problem related to ${\mathcal{V}}_n$ is equivalent to Connes' Embedding problem.

arXiv comment29 pages, final version to appear in the Indiana University Mathematics Journal