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 : t_j = t_j^*, 0 < t_j ≤ 1 for 1 ≤ j ≤ 2⟩ and positive, invertible contractions x₁,x₂ in a finite von Neumann algebra M with trace τ such that τ(p(x₁,x₂)) < 0 and Tr_k(p(A₁,A₂)) ≥ 0 for every positive integer k and all positive definite contractions A₁,A₂ 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_i² 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 t_i is nonnegative or both of the coefficients of t₁ and t₂ are negative.

Keywords

Mathematical Subject Classification 2000

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