A noncommutative moment problem (Q5890185)

Let \({\mathcal A}_n\) be the universal unital \(C^*\)-algebra generated by elements \(t_1,\cdots,t_n\) subject to \(-1\leq t_j=t_j ^*\leq 1\), that is, the \(C^*\)-algebraic free product of \(n\) copies of \(C[-1,1]\). Let \({\mathbf P}_n\) be the space of noncommutative polynomials in \(t_1,\cdots,t_n\). A linear functional \(\phi\) on \({\mathbf P}_n\) is called tracial if \(\phi(pq)=\phi(qp)\) for all \(p,q\in {\mathbf P}_n\). NEWLINENEWLINENEWLINEThe author proves that a tracial linear functional \(\phi\) on \({\mathbf P}_n\) can be extended to a tracial positive unital linear functional on \({\mathcal A}_n\) if and only if (1) \(\phi(1)=1\), (2) \(\lim\inf_{m\to\infty} |\phi(t_j^{2m})|<\infty\) and (3) \(\phi(p^*p)\geq 0\) for every \(p\in{\mathbf P}_n\). For the one variable case this statement coincides with the solvability criterion of the Hausdorff moment problem. NEWLINENEWLINENEWLINEThen the author relates this result to the \textit{A. Connes}' problem [Ann. Math., II. Ser. 104, 73-115 (1976; Zbl 0343.46042)] of embedding finite factor von Neumann algebras into an ultraproduct of the hyperfinite \(\text{II}_{1}\) factor. It is shown that the Connes' problem is equivalent to characterizing of noncommutative polynomials which have positive trace, whenever all variables are replaced by contractive Hermitian matrices.