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Göteborgs universitets publikationer

Trace-positive complex polynomials in three unitaries

Författare och institution:
Stanislav Popovych (Institutionen för matematiska vetenskaper, Chalmers/GU)
Publicerad i:
Proceedings of the American Mathematical Society, 138 ( 10 ) s. 3541-3550
Artikel, refereegranskad vetenskaplig
Sammanfattning (abstract):
We consider the quadratic polynomials in three unitary generators, i.e. the elements of the group *-algebra of the free group with generators u2, u3 of the form f = Sigma(3)(j,k=1) alpha(jk)u(j)(*)u(k), alpha(jk) is an element of C We prove that if f is self-adjoint and Tr(f(U-1, U-2, U-3)) >= 0 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 operatorsU(1),..., U-4 connected by the linear relation a(1)U(1) + a(2)U(2) + a(3)U(3) + a(4)U(4) = 0.
Ämne (baseras på Högskoleverkets indelning av forskningsämnen):
Matematik ->
Beräkningsmatematik ->
Tillämpad matematik
Connes' Embedding Conjecture, II1-factor, trace, sum of hermitian, squares, Tarski sentence, discriminant variety, asterisk-algebras, schubert calculus, matrices, eigenvalues, extensions, products
Postens nummer:
Posten skapad:
2010-10-26 10:20
Posten ändrad:
2012-03-08 11:47

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