Projective Schur groups of Henselian fields

One of the open questions that has emerged in the study of the projective Schur group PS(F) of a field FF is whether or not PS(F) is an algebraic relative Brauer group over FF, i.e. does there exist an algebraic extension L/FL/F such that PS(F)=Br(L/F)? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that PS(F) is an algebraic relative Brauer group for all Henselian valued fields FF of equal characteristic whose residue field is a local or global field. For this, we first show how PS(F) is determined by PS(k) for an equicharacteristic Henselian field with arbitrary residue field kk.