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

Abstract

The derivation constant $K(A)\ge \frac{1}{2}$ has been extensively studied for \emph{unital} non-commutative $C^{\ast }$-algebras. In this paper, we investigate properties of $K(M(A))$ where $M(A)$ is the multiplier algebra of a non-unital $C^{\ast }$-algebra $A$. A number of general results are obtained which are then applied to the group $C^{\ast }$-algebras $A=C^{\ast }(G_N)$ where $G_N$ is the motion group ${\mathbb{R}}^N⋊SO(N)$. Utilising the rich topological structure of the unitary dual $\widehat{G_N}$, it is shown that, for $N\ge 3$, $$K(M(C^{\ast }(G_N)))=\frac{1}{2}\lceil \frac{N}{2}\rceil .$$

26 pages. This is an Author Accepted Manuscript. Publication details will be added when available

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