Norms of inner derivations for multiplier algebras of C⁎-algebras and group C⁎-algebras

The derivation constant has been previously studied for unital non-commutative C⁎-algebras A. This paper begins the study of K(M(A)) where M(A) is the multiplier algebra of a non-unital C⁎-algebra A. Two results are obtained giving separate conditions on A which imply that K(M(A))⩽1. These results are applied to A=C⁎(G) for a number of locally compact groups G including SL(2,R), SL(2,C) and several 2-step solvable groups. In these cases, K(M(A))=1. On the other hand, if G is a (non-abelian) amenable [SIN]-group then .