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

Abstract The derivation constant K ( A ) ≥ 1 2 has been extensively studied for unital non-commutative C ⁎ -algebras. In this paper, we investigate properties of K ( M ( A ) ) where M ( A ) is the multiplier algebra of a non-unital C ⁎ -algebra A . A number of general results are obtained which are then applied to the group C ⁎ -algebras A = C ⁎ ( G N ) where G N is the motion group R N ⋊ SO ( N ) . Utilizing the rich topological structure of the unitary dual G N ˆ , it is shown that, for N ≥ 3 , K ( M ( C ⁎ ( G N ) ) ) = 1 2 ⌈ N 2 ⌉ .