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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

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 < tj 1 for 1 j 2 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 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.

von Neumann algebras, noncommutative moment problems, Connes embedding conjecture
Mathematical Subject Classification 2000
Primary: 46L10
Secondary: 46L54
Received: 9 July 2010
Accepted: 26 February 2011
Published: 22 September 2011

Communicated by David R. Larson
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