Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2010, January 18.

Abstract:
We present tracial analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in non-commuting variables with values invariant under cyclic permutations of the indexes, is called a tracial sequence. We prove that such a sequence can be represented with tracial moments of matrices if its corresponding moment matrix is positive semidefinite and of finite rank. A truncated tracial sequence allows for such a representation if and only if one of its extensions admits a flat extension. Finally, we apply the theory via duality to investigate trace-positive polynomials in non-commuting variables.

Mathematics Subject Classification (2000): 47A57, 15A45, 13J30, 08B20, 11E25, 44A60.

Keywords and Phrases: (truncated) moment problem, non-commutative polynomial, sum of hermitian squares, moment matrix, free positivity.

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