Representations of C*-algebras in dual and right dual Banach algebras

The range of a contractive algebra morphism from a C*-algebra to a Banach algebra is closed, and the morphism is a C*-morphism onto its range. When the codomain is a dual Banach algebra, or only a right dual Banach algebra, such a morphism extends to a W*-morphism onto the weak star closure of the range (at least in the unital case). Boolean algebras of contractive projections (in right dual Banach algebras) have weak star completions; and operators with a contractive functional calculus on a dual Banach space are scalar type prespectral. Some of these results extend to morphisms that are neither unital nor contractive, so long as one can renorm the codomain dually in a suitable manner, as when the codomain is a dual Banach algebra, or when the range of the natural extension of the morphism forms a commuting set