A note on moments in finite von Neumann algebras

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 $\mathcal{A}=\langle t_1,t_2:t_j=t_j^{\ast },0<t_j\le 1for1\le j\le 2\rangle$ and positive, invertible contractions $x_1,x_2$ in a finite von Neumann algebra $\mathcal{ℳ}$ 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 $p\in \mathcal{A}$ 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.