Unitary SK1 of graded and valued division algebras

The reduced unitary Whitehead group SK1 of a graded division algebra equipped with a unitary involution (that is, an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their non-graded counterparts. The bridge to the non-graded case is established by proving that the unitary SK1 of a tame-valued division algebra with a unitary involution over a henselian field coincides with the unitary SK1 of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary SK1 for much wider classes of division algebras over henselian fields.