Tracepositive complex polynomials in three unitaries
Author:
Stanislav Popovych
Journal:
Proc. Amer. Math. Soc. 138 (2010), 35413550
MSC (2000):
Primary 46L10; Secondary 15A48
Published electronically:
June 4, 2010
MathSciNet review:
2661554
Fulltext PDF
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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 of the form , . We prove that if is selfadjoint 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, SE412 96 Göteborg, Sweden
Email:
popovych@univ.kiev.ua
DOI:
http://dx.doi.org/10.1090/S000299392010103143
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.
