An Explicit Polynomial Counterexample to Connes'Embedding Conjecture

Abstract

We construct an explicit Hermitian polynomial in six selfadjoint variables, $f=-1+\omega+\omega^*+M\sum_{j=1}^{15}(2-u_j-u_j^*)$, with integer coefficients, degree $72$, and exactly $33$ monomials. Here $\omega,u_1,\ldots,u_{15}$ are specified words, $*$ reverses words, and $M$ is a specified positive integer. Its normalized trace is at least $3/4$ on selfadjoint matrix contraction tuples of every dimension, but equals $-1$ at a specified tuple of selfadjoint unitaries in a group von Neumann algebra. Thus $f$ is a counterexample to the algebraic formulation of Connes'embedding conjecture. We also construct a Hermitian quartic in $37$ selfadjoint variables, with coefficients in $\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}$ and $387$ monomials, attaining the same trace bounds without norm restrictions on selfadjoint matrix inputs. Degree four is minimal in this unrestricted setting. Finally, an encoding in two selfadjoint variables yields a counterexample on the contraction domain with integer coefficients and degree at most $48$; two variables are minimal.