Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Non-Commutative Unitary Analogue of Kirchberg's Conjecture   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.

11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph