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A note on moments in finite von Neumann algebras. (English) Zbl 1238.46049

Summary: By a result of the second author [Proc. Am. Math. Soc. 129, No. 6, 1785–1791 (2001; Zbl 0982.44005)], the Connes embedding conjecture (CEC) is false if and only if there exists a self-adjoint noncommutative polynomial \(p(t_1,t_2)\) in the universal unital \(C^{\ast}\)-algebra \(\mathcal A = \langle t_1,t_2 : t_j = t_j^\ast,0<t_j\leq1\text{ for }1\leq j\leq2\rangle\) and positive, invertible contractions \(x_1,x_2\) in a finite von Neumann algebra \(\mathcal M\) with trace \(\tau\) such that \(\tau(p(x_1,x_2))<0\) and \(\text{Tr}_k(p(A_1,A_2)) \geq 0\) for every positive integer \(k\) and all positive definite contractions \(A_1,A_2\) in \(M_k(\mathbb C)\). 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.

MSC:

46L10 General theory of von Neumann algebras
46L54 Free probability and free operator algebras
46L36 Classification of factors

Citations:

Zbl 0982.44005
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