Trace-positive quadratic polynomials and SOS decompositions in free group algebra

We present reformulation of Connes' Embedding Conjecture about approximation of the trace on II_1 von Neumann algebra by matrix traces in terms of SOS decompositions of trace-positive quadratic non-commutative polynomials in unitary generators. We discuss the possibility of such decompositions in the case of real coefficients and cases of polynomilas with small number of generators.