Degree and arity of tracial separation for self-adjoint contractions

Abstract

Motivated by the recent explicit counterexample to Connes' embedding conjecture, we study the degree and number of variables needed to separate normalized matrix traces on self-adjoint contractions from traces on finite von Neumann algebras. We prove that the two trace infima coincide for every Hermitian polynomial of degree at most two in at most three variables, with a dimension-explicit approximation bound. We give three explicit separators: an integer-coefficient polynomial of degree six in ten variables with 392 nonzero words, a quadratic polynomial in 214 variables, and an integer-coefficient polynomial in two variables of ordinary and cyclic degree 24. Each has normalized trace at least \(3/4\) on every matrix contraction tuple and trace \(-1\) at a finite tracial witness. We provide the source systems, transfer estimates, and exact coefficient recipes for all three constructions.