Trace-positive complex polynomials in three unitaries

Author:
Stanislav Popovych

Journal:
Proc. Amer. Math. Soc. **138** (2010), 3541-3550

MSC (2000):
Primary 46L10; Secondary 15A48

DOI:
https://doi.org/10.1090/S0002-9939-2010-10314-3

Published electronically:
June 4, 2010

MathSciNet review:
2661554

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the quadratic polynomials in three unitary generators, i.e. the elements of the group -algebra of the free group with generators of the form , . We prove that if is self-adjoint and for arbitrary unitary matrices , then is a sum of hermitian squares. To prove this statement we reduce it to the question whether a certain Tarski sentence is true. Tarski's decidability theorem thus provides an algorithm to answer this question. We use an algorithm due to Lazard and Rouillier for computing the number of real roots of a parametric system of polynomial equations and inequalities implemented in Maple to check that the Tarski sentence is true.

As an application, we describe the set of parameters such that there are unitary operators connected by the linear relation .

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Additional Information

**Stanislav Popovych**

Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Göteborg, Sweden

Email:
popovych@univ.kiev.ua

DOI:
https://doi.org/10.1090/S0002-9939-2010-10314-3

Keywords:
Connes' Embedding Conjecture,
$II_{1}$-factor,
trace,
sum of hermitian squares,
Tarski sentence,
discriminant variety.

Received by editor(s):
January 6, 2009

Received by editor(s) in revised form:
November 11, 2009

Published electronically:
June 4, 2010

Communicated by:
Marius Junge

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.