# Journal of Operator Theory

Volume 68, Issue 1, Summer 2012 pp. 141-163.

The truncated tracial moment problem**Authors**: Sabine Burgdorf (1) and Igor Klep (2)

**Author institution:**(1) SB -- MATHGEOM -- EGG, EPFL, Station 8, 1015 Lausanne, Suisse

(2) Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

**Summary:**We present \textit{tracial} analogs of the classical results of Curto and Fialkow on moment matrices. A sequence of real numbers indexed by words in noncommuting variables with values invariant under cyclic permutations of the indexes, is called a \textit{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 \textit{truncated} tracial sequence allows for such a representation if and only if one of its extensions admits a flat extension. Finally, we apply this theory via duality to investigate trace-positive polynomials in noncommuting variables.

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