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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: We consider the quadratic polynomials in three unitary generators, i.e. the elements of the group $ *$-algebra of the free group with generators $ u_1, u_2, u_3$ of the form $ f=\sum_{j, k=1}^{3}\alpha_{jk}u_{j}^{*}u_{k}$, $ \alpha_{jk} \in \mathbb{C}$. We prove that if $ f$ is self-adjoint and $ {Tr}(f(U_{1}, U_2 ,U_{3}))\ge0$ for arbitrary unitary matrices $ U_{1}, U_2, U_3$, then $ f$ 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 $ a_1, a_2, a_3, a_4$ such that there are unitary operators $ U_1, \ldots, U_4$ connected by the linear relation $ a_1 U_1+a_2 U_2 +a_3 U_3 +a_4 U_4 =0$.


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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.

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