#### Vol. 4, No. 1, 2011

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A note on moments in finite von Neumann algebras

### Jon Bannon, Donald Hadwin and Maureen Jeffery

Vol. 4 (2011), No. 1, 65–74
##### Abstract

By a result of the second author, the Connes embedding conjecture (CEC) is false if and only if there exists a self-adjoint noncommutative polynomial $p\left({t}_{1},{t}_{2}\right)$ in the universal unital ${C}^{\ast }$-algebra $\mathsc{A}=〈{t}_{1},{t}_{2}:{t}_{j}={t}_{j}^{\ast }\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}0<{t}_{j}\le 1for1\le j\le 2〉$ and positive, invertible contractions ${x}_{1},{x}_{2}$ in a finite von Neumann algebra $\mathsc{ℳ}$ with trace $\tau$ such that $\tau \left(p\left({x}_{1},{x}_{2}\right)\right)<0$ and ${Tr}_{k}\left(p\left({A}_{1},{A}_{2}\right)\right)\ge 0$ for every positive integer $k$ and all positive definite contractions ${A}_{1},{A}_{2}$ in ${M}_{k}\left(ℂ\right)$. We prove that if the real parts of all coefficients but the constant coefficient of a self-adjoint polynomial $p\in \mathsc{A}$ have the same sign, then such a $p$ cannot disprove CEC if the degree of $p$ is less than $6$, and that if at least two of these signs differ, the degree of $p$ is $2$, the coefficient of one of the ${t}_{i}^{2}$ is nonnegative and the real part of the coefficient of ${t}_{1}{t}_{2}$ is zero then such a $p$ disproves CEC only if either the coefficient of the corresponding linear term ${t}_{i}$ is nonnegative or both of the coefficients of ${t}_{1}$ and ${t}_{2}$ are negative.

##### Keywords
von Neumann algebras, noncommutative moment problems, Connes embedding conjecture
Primary: 46L10
Secondary: 46L54