Norms of inner derivations for multiplier algebras of \(C^*\)-algebras and group \(C^*\)-algebras
Let \(A\) be a C\(^*\)-algebra. The derivation constant \(K(A)\) is the smallest number in \([0,\infty]\) such that \[ K(A)\|D(a,A)\|\geq d(a,Z(A)),\quad (a\in A), \] where \(D(a,A)\) is the inner derivation generated by \(a\) and \(d(a,Z(A))\) is the distance from \(a\) to \(Z(A)\). If we consider self-adjoint \(a\in A\), then the corresponding constant is denoted by \(K_s(A)\). Note that it is well known that \[ \|D(a,A)\|\leq 2d(a,Z(A))\quad (a\in A). \] It follows from this that \(K(A)\geq 1/2\) for unital non-commutative C\(^*\)-algebras \(A\). Also it is known that \(K(A)= 1/2\) for a unital primitive C\(^*\)-algebra or von Neumann algebra; see [\textit{J. G. Stampfli}, Pac. J. Math. 33, 737--747 (1970; Zbl 0197.10501)] and [\textit{L. Zsidó}, Proc. Am. Math. Soc. 38, 147--150 (1973; Zbl 0257.46103)]. The authors in this paper study the derivation constant of \(M(A)\), the multiplier algebra of a non-unital C\(^*\)-algebra \(A\). They mention that for many non-unital C\(^*\)-algebras \(A\) the constant \(K(A)\) provides less information than \(K(M(A))\). Let \(G\) be a locally compact amenable group. They show that \(C^*(G)\) is quasi-standard and apply the results of Section 3 of the paper and show that \(K(M(C^*(G)))\leq 1\) and for a second countable group \(G\) they show that \(K(M(C^*(G)))=K_s(M(C^*(G)))= 1\). In particular they consider \(C^*(G)\) for a number of locally compact groups \(G\) including \(SL(2,\mathbb R),\,\, SL(2,\mathbb C)\) and \(\mathbb R^2\rtimes so(2)\). Also for non-abelian amenable SIN-groups \(G\) or second countable, type I and non-abelian groups they show that \(K(M(C^*(G)))=K_s(M(C^*(G)))= 1/2\).
scientific article; zbMATH DE number 6017638
scientific article