An Explicit Polynomial Counterexample to Connes’ Embedding Conjecture
We construct an explicit Hermitian polynomial $f$ with integer coefficients, of degree $12$ in $65$ selfadjoint variables, whose normalized trace is at least $3/4$ on every tuple of selfadjoint matrix contractions, in every dimension, and equals $-1$ at a specified tuple of selfadjoint unitaries in a group von Neumann algebra. Consequently, $f+\varepsilon$ lies outside the contraction quadratic module modulo commutators for $0\le\varepsilon<1$, giving an explicit counterexample to the algebraic formulation of Connes’ embedding conjecture.
Combining the group construction of Kun and Thom with the normalization argument of Thom and the spectral correction theorem of Alekseev, Liu, and Thom, we determine an explicit positive integer $\mu$ for which
\[f=1-(P-Q)^2+\mu\sum_{\nu=1}^{825}E_\nu^*E_\nu+\mu\sum_{j=1}^{65}(1-X_j^2)^2.\]Here $P,Q$ encode conjugate involutions, and the $E_\nu$ encode relation defects.
