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

Full-text: Open access

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<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

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

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

Permanent link to this document
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

Keywords
von Neumann algebras noncommutative moment problems Connes embedding conjecture

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


Export citation

References

  • M. Dostál and D. Hadwin, “An alternative to free entropy for free group factors”, Acta Math. Sin. $($Engl. Ser.$)$ 19:3 (2003), 419–472.
  • D. Hadwin, “A noncommutative moment problem”, Proc. Amer. Math. Soc. 129:6 (2001), 1785–1791.
  • R. V. Kadison and J. R. Ringrose, Fundamentals of the theory of operator algebras, I: Elementary theory, Pure and Applied Mathematics 100-I, Academic Press, New York, 1983.
  • F. Rădulescu, “Convex sets associated with von Neumann algebras and Connes' approximate embedding problem”, Math. Res. Lett. 6:2 (1999), 229–236.
  • A. M. Sinclair and R. R. Smith, Finite von Neumann algebras and masas, London Math. Soc. Lecture Note Series 351, Cambridge University Press, 2008.
  • D. Voiculescu, “The analogues of entropy and of Fisher's information measure in free probability theory, II”, Invent. Math. 118:3 (1994), 411–440.