## Involve: A Journal of Mathematics

• Involve
• Volume 4, Number 1 (2011), 65-74.

### 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 $p(t1,t2)$ in the universal unital $C∗$-algebra $A=〈t1,t2:tj=tj∗,0 and positive, invertible contractions $x1,x2$ in a finite von Neumann algebra $ℳ$ with trace $τ$ such that $τ(p(x1,x2))<0$ and $Trk(p(A1,A2))≥0$ for every positive integer $k$ and all positive definite contractions $A1,A2$ in $Mk(ℂ)$. We prove that if the real parts of all coefficients but the constant coefficient of a self-adjoint polynomial $p∈A$ 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 $ti2$ is nonnegative and the real part of the coefficient of $t1t2$ is zero then such a $p$ disproves CEC only if either the coefficient of the corresponding linear term $ti$ is nonnegative or both of the coefficients of $t1$ and $t2$ are negative.

#### Article information

Source
Involve, Volume 4, Number 1 (2011), 65-74.

Dates
Accepted: 26 February 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513733363

Digital Object Identifier
doi:10.2140/involve.2011.4.65

Mathematical Reviews number (MathSciNet)
MR2838262

Zentralblatt MATH identifier
1238.46049

Subjects
Primary: 46L10: General theory of von Neumann algebras
Secondary: 46L54: Free probability and free operator algebras

#### Citation

Bannon, Jon; Hadwin, Donald; Jeffery, Maureen. A note on moments in finite von Neumann algebras. Involve 4 (2011), no. 1, 65--74. doi:10.2140/involve.2011.4.65. https://projecteuclid.org/euclid.involve/1513733363

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