A note on moments in finite von Neumann algebras

Bannon, Jon; Hadwin, Donald; Jeffery, Maureen

doi:10.2140/involve.2011.4.65

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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

DOI: 10.2140/involve.2011.4.65

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 $p (t_{1}, t_{2})$ in the universal unital $C^{*}$ -algebra $A = 〈 t_{1}, t_{2} : t_{j} = t_{j}^{*}, 0 < t_{j} \leq 1 f o r 1 \leq j \leq 2 〉$ and positive, invertible contractions $x_{1}, x_{2}$ in a finite von Neumann algebra $ℳ$ with trace $τ$ such that $τ (p (x_{1}, x_{2})) < 0$ and ${Tr}_{k} (p (A_{1}, A_{2})) \geq 0$ for every positive integer $k$ and all positive definite contractions $A_{1}, A_{2}$ in $M_{k} (ℂ)$ . We prove that if the real parts of all coefficients but the constant coefficient of a self-adjoint polynomial $p \in A$ 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

Mathematical Subject Classification 2000

Primary: 46L10

Secondary: 46L54

Milestones

Received: 9 July 2010

Accepted: 26 February 2011

Published: 22 September 2011

Communicated by David R. Larson

Authors

Jon Bannon
Department of Mathematics Siena College Loudonville, NY 12211 United States

Donald Hadwin
Department of Mathematics and Statistics The University of New Hampshire Durham, NH 03824 United States

Maureen Jeffery
Department of Mathematics Siena College Loudonville, NY 12211 United States

Vol. 4, No. 1, 2011