A Positivstellensatz on the Matrix Algebra of Finitely Generated Free Group

Positivstellens\"atze are a group of theorems on the positivity of involution algebras over R or C. One of the most well-known Positivstellensatz is the solution to Hilbert's 17th problem given by E. Artin, which asserts that a real polynomial in n commutative variables is nonnegative on real affine space if and only if it is a sum of fractional squares. Let m and n be two positive integers. For the free group Fₙ generated by n letters, and a symmetric polynomial b with variables in Fₙ and with n-by-n complex matrices coefficients, we use real algebraic geometry to give a new proof showing that b is a sum of Hermitian squares if and only if b is mapped to a positive semidefinite matrix under any finitely dimensional unitary representation of Fₙ.