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

The derivation constant K(A) 12 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 K(M(A)) = 12 . © 2011 Elsevier Inc. All rights reserved.