Involve: A Journal of Mathematics

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

A note on moments in finite von Neumann algebras

Jon Bannon, Donald Hadwin, and Maureen Jeffery

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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<tj1for1j2 and positive, invertible contractions x1,x2 in a finite von Neumann algebra 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(). We prove that if the real parts of all coefficients but the constant coefficient of a self-adjoint polynomial pA 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

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

Received: 9 July 2010
Accepted: 26 February 2011
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

von Neumann algebras noncommutative moment problems Connes embedding conjecture


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.

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