# A non-commutative, analytic version of Hilbert's 17-th problem in type II$_1$ von Neumann algebras

@article{Radulescu2004ANA, title={A non-commutative, analytic version of Hilbert's 17-th problem in type II\$\_1\$ von Neumann algebras}, author={Florin Radulescu}, journal={arXiv: Operator Algebras}, year={2004} }

We prove a non-commutative version of the Hilbert's 17th problem, giving a characterization of the class of non-commutative polynomials in n-undeterminates that have positive trace when evaluated in n-selfadjoint elements in arbitrary II1 von Neumann algebra. As a corollary we prove that Connes's embedding conjecture is equivalent to a statement that can be formulated entirely in the context of finite matrices.

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