Trace-positive polynomials

Igor Klep

Vol. 250 (2011), No. 2, 339–352

DOI: 10.2140/pjm.2011.250.339

Abstract

In this paper positivity of polynomials in free noncommuting variables in a dimension-dependent setting is considered. That is, the images of a polynomial under finite-dimensional representations of a fixed dimension are investigated. It is shown that unlike in the dimension-free case, every trace-positive polynomial is (after multiplication with a suitable denominator—a Hermitian square of a central polynomial) a sum of a positive semidefinite polynomial and commutators. Together with our previous results this yields the following Positivstellensatz: every trace-positive polynomial is modulo sums of commutators and polynomial identities a sum of Hermitian squares with weights and denominators. Understanding trace-positive polynomials is one of the approaches to Connes’ embedding conjecture.

Keywords

free algebra, noncommutative polynomial, central simple algebra, (reduced) trace, polynomial identity, involution, central polynomial, quadratic form, free positivity

Mathematical Subject Classification 2000

Primary: 16W10, 13J30

Secondary: 11E25, 16R50

Milestones

Received: 31 December 2009

Accepted: 15 September 2010

Published: 31 March 2011