Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Published electronically: June 4, 2010
MathSciNet review: 2661554
Full-text PDF

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 $ 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$.

References [Enhancements On Off] (What's this?)

  • 1. S. Agnihotri, C. Woodward, Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Research Letters 5 (1998), 817-836. MR 1671192 (2000a:14066)
  • 2. M. Bakonyi, D. Timotin, Extensions of positive definite functions on free groups, J. Funct. Anal. 246 (2007), 31-49. MR 2316876 (2008g:43009)
  • 3. S. Corvez, F. Rouillier, Using computer algebra tools to classify serial manipulators, in Automated Deduction in Geometry. Lecture Notes in Comput. Sci., vol. 2930. Springer, 2004, pp. 31-43. MR 2090401
  • 4. K. Dykema, K. Juschenko, Matrices of unitary moments, arXiv:0901.0288.
  • 5. W. Fulton, Eigenvalues, invariant factors, highest weights and Schubert calculus, Bull. Amer. Math. Soc. 37 (2000), 209-249. MR 1754641 (2001g:15023)
  • 6. K. Juschenko, S. Popovych, Algebraic reformulation of Connes' embedding problem and the free group algebra, Israel J. Math, in press.
  • 7. E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of group $ C^{*}$-algebras, Invent. Math. 112 (1993), no. 3, 449-489. MR 1218321 (94d:46058)
  • 8. I. Klep, M. Schweighofer, Connes' embedding conjecture and sums of Hermitian squares, Adv. Math. 217 (2008), 1816-1837. MR 2382741 (2009g:46109)
  • 9. D. Lazard, F. Rouillier, Solving parametric polynomial systems, J. Symbolic Comput. 42 (2007), 636-667. MR 2325919 (2008g:68131)
  • 10. S. McCullough, Factorization of operator valid polynomials in several non-commuting variables, Linear Algebra Appl. 326 (2001), 193-203. MR 1815959 (2002f:47035)
  • 11. D. Mumford, The red book of varieties and schemes. In Lecture Notes in Mathematics, vol. 1358, Springer-Verlag, 1988. MR 971985 (89k:14001)
  • 12. S. Popovych, Positive semidefinite quadratic forms on unitary matrices, Linear Algebra Appl. 433 (2010), no. 1, 164-171.
  • 13. S. Popovych, On $ O^*$-representability and $ C^*$-representability of $ *$-algebras, Houston J. Math. 36 (2010), no. 2, 591-617.
  • 14. A. Tarski, A Decision Method for Elementary Algebra and Geometry, University of California Press, 1951. MR 0044472 (13:423a)
  • 15. B. Wang, F. Zhang, A trace inequality for unitary matrices, Amer. Math. Monthly 101 (1994), no. 5, 453-455. MR 1272947 (95a:15015)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L10, 15A48

Retrieve articles in all journals with MSC (2000): 46L10, 15A48

Additional Information

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

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.

American Mathematical Society