This article studies algebraic certificates of positivity for noncommutative (nc) operator-valued polynomials on matrix convex sets, such as the solution set D L , called a free Hilbert spectrahedron, of the linear operator inequality (LOI) L(X) = A 0 ⊗ I + g j=1 A j ⊗ X j 0, where A j are self-adjoint linear operators on a separable Hilbert space, X j matrices and I is an identity matrix. If A j are matrices, then L(X) 0 is called a linear matrix inequality (LMI) and D L a free spectrahedron. For monic LMIs, i.e., A 0 = I, and nc matrix-valued polynomials the certificates of positivity were established by Helton, Klep and McCullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry. Since the full strength of the theory of complete positivity is not restricted to finite dimensions, but works well also in the infinite-dimensional setting, we use it to tackle our problems. First we extend the characterization of the inclusion D L 1 ⊆ D L 2 from monic LMIs to monic LOIs L 1 and L 2 . As a corollary one immediately obtains the description of a polar dual of a free Hilbert spectrahedron D L and its projection, called a free Hilbert spectrahedrop. Further on, using this characterization in a separation argument, we obtain a certificate for multivariate matrix-valued nc polynomials F positive semidefinite on a free Hilbert spectrahedron defined by a monic LOI. Replacing the separation argument by an operator Fejér-Riesz theorem enables us to extend this certificate, in the univariate case, to operator-valued polynomials F . Finally, focusing on the algebraic description of the equality D L 1 = D L 2 , we remove the assumption of boundedness from the description in the LMIs case by an extended analysis. However, the description does not extend to LOIs case by counterexamples.

We extend Haviland's theorem on the integral representation of positive linear functionals on usual (real multivariate) polynomials to the integral representation of positive linear maps on operator polynomials mapping into the space of operators. We also extend its special case of compact semialgebraic set studied by Schm\"udgen to matrix polynomials and show with a counterexample, that the extension to the operator polynomials fails in general

A * -linear map Φ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations I n ⊗ Φ are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.Date: February 22, 2018. 2010 Mathematics Subject Classification. 13J30, 46L07, 52A40 (Primary); 47L25, 81P45, 90C22 (Secondary).

Abstract. Regular normalized W -valued spectral measures on a compact Hausdorff space X are in one-to-one correspondence with unital * -representations ρ : C(X, C) → W , where W stands for a von Neumann algebra. In this paper we show that for every compact Hausdorff space X and every von Neumann algebras W 1 , W 2 there is a one-to-one correspondence between unital * -representations ρ : C(X, W 1 ) → W 2 and special B(W 1 , W 2 )-valued measures on X that we call non-negative spectral measures. Such measures are special cases of non-negative measures that we introduced in our previous paper [3] in connection with moment problems for operator polynomials.

The (classical) truncated moment problem, extensively studied by Curto and Fialkow, asks to characterize when a finite sequence of real numbers indexes by words in commuting variables can be represented with moments of a positive Borel measure µ on R n . In [BK12] Burgdorf and Klep introduced its tracial analog, the truncated tracial moment problem, which replaces commuting variables with non-commuting ones and moments of µ with tracial moments of matrices. In the bivariate quartic case, where indices run over words in two variables of degree at most four, every sequence with a positive definite 7 × 7 moment matrix M 2 can be represented with tracial moments [BK10,BK12]. In this article the case of singular M 2 is studied. For M 2 of rank at most 5 the problem is solved completely; namely, concrete measures are obtained whenever they exist and the uniqueness question of the minimal measures is answered. For M 2 of rank 6 the problem splits into four cases, in two of which it is equivalent to the feasibility problem of certain linear matrix inequalities. Finally, the question of a flat extension of the moment matrix M 2 is addressed. While this is the most powerful tool for solving the classical case, it is shown here by examples that, while sufficient, flat extensions are mostly not a necessary condition for the existence of a measure in the tracial case. INTRODUCTION1.1. Context. The Moment problem (MP) is a classical question in analysis and concerns the existence of a positive Borel measure µ supported on a subset K of R n , representing a given sequence of real or complex numbers indexed by monomials as the integration of the corresponding monomials w.r.t. µ; nice expositions on the MP are [Akh65, KN77]. The solution to the MP on R n is given by Haviland's theorem [Hav35], which establishes the duality with positive polynomials and relates the MP to real algebraic geometry (RAG). One of the cornerstones of RAG is the celebrated Schmüdgen theorem [Sch91], which solves the problem on compact basic closed semialgebraic sets and is the beginning of extensive research of the MP in RAG; we refer the reader to [Put93, PV99, DP01, PS01, KM02, Lau05, Lau09, Mar08, Las09] and the references therein for further details. Another important aspect of the MP is uniqueness of the representing measures. For compact sets the measure is unique (see e.g., [Mar08]), while for noncompact sets, the question of uniqueness is highly nontrivial (see [PS06,PS08]). There are various generalizations of the MP. Functional analysis studies various versions of matrix and operator MPs; see [Kre49, Kov83, AV03, Vas03, BW11, CZ12, KW13] and references therein. The quantum MP from quantum physics is considered in [DLTW08]. The rational MP, which extends Schmüdgen theorem from the polynomial algebra to its localizations, is solved in [CMN11], while [GKM16] investigates the MP for the polynomial algebra in infinitely many variables. The MP on semialgebraic sets of generalized function is considered in [IKR14]. The beginning of free RAG is the sol...