The truncated tracial moment problem

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.

[1]  Sterling K. Berberian,et al.  Lectures in functional analysis and operator theory , 1974 .

[2]  E. Haviland,et al.  On the Momentum Problem for Distribution Functions in More Than One Dimension. II , 1935 .

[3]  Stefan Rolewicz,et al.  On a problem of moments , 1968 .

[4]  Jean-Pierre Tignol,et al.  The Book of Involutions , 1998 .

[5]  Igor Klep,et al.  Connes' embedding conjecture and sums of hermitian squares , 2008 .

[6]  Mihai Putinar,et al.  Noncommutative sums of squares , 2005 .

[7]  D. Bessis,et al.  Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics , 1975 .

[8]  A. Connes,et al.  Classification of Injective Factors Cases II 1 , II ∞ , III λ , λ 1 , 1976 .

[9]  J. Helton.  “Positive” noncommutative polynomials are sums of squares , 2002 .

[10]  Theworkof Alain Connes.  CLASSIFICATION OF INJECTIVE FACTORS , 1981 .

[11]  Charles N. Delzell,et al.  Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra , 2001 .

[12]  Alexander Barvinok,et al.  A course in convexity , 2002, Graduate studies in mathematics.

[13]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[14]  K. Schmüdgen.  TheK-moment problem for compact semi-algebraic sets , 1991 .

[15]  Tsit Yuen Lam,et al.  A first course in noncommutative rings , 2002 .

[16]  R. Curto,et al.  Flat Extensions of Positive Moment Matrices: Recursively Generated Relations , 1998 .

[17]  Victoria Powers,et al.  The moment problem for non-compact semialgebraic sets , 2001 .

[18]  Igor Klep,et al.  Sums of Hermitian Squares and the BMV Conjecture , 2008 .

[19]  Scott McCullough.  Factorization of operator-valued polynomials in several non-commuting variables☆ , 2001 .

[20]  K. Schmüdgen.  TheK-moment problem for compact semi-algebraic sets , 1991 .

[21]  Salma Kuhlmann,et al.  Positivity, sums of squares and the multi-dimensional moment problem , 2002 .

[22]  Raúl E. Curto,et al.  Solution of the Truncated Complex Moment Problem for Flat Data , 1996 .

[23]  Mihai Putinar,et al.  Solving moment problems by dimensional extension , 1999, math/9905215.

[24]  M. Marshall.  Positive polynomials and sums of squares , 2008 .

[25]  Jiawang Nie,et al.  Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems , 2009, 0908.3230.

[26]  M. Laurent.  Sums of Squares, Moment Matrices and Optimization Over Polynomials , 2009 .

[27]  Monique Laurent,et al.  Revisiting two theorems of Curto and Fialkow on moment matrices , 2005 .