On *-clean non-commutative group rings

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups D2n, and the generalized quaternion groups Q2n with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of RD2pk with a prime p ∈ J(R), and RD2n with 2 ∈ J(R), where R is a commutative local ring. For the semisimple group algebra case, we investigate when KG is ∗-clean, where K is the field of rational numbers ℚ or a finite field Fq and G = D2n or G = Q2n.