Derivations on C*-Algebras: A Revisit

The most basic class of derivations on C*-algebras consists of the inner derivations—those expressible as commutators with elements from the multiplier algebra. A fundamental question in the theory of C*-algebras is to determine which algebras admit only inner derivations. Landmark results by Sakai, Kadison, and Sproston established this property for all von Neumann algebras, simple C*-algebras, and homogeneous C*-algebras. In the separable setting, the problem was completely resolved in 1979 by Akemann, Elliott, Pedersen, and Tomiyama, who showed that a separable C*-algebra has only inner derivations if and only if it is a direct sum of a C*-algebra with continuous trace and a C*-algebra with discrete primitive spectrum. However, the non-separable case remains largely unsettled—even for 2-subhomogeneous algebras. In 1978, Pedersen posed a unifying question, inspired by the work of Sakai and Kadison: given a C*-algebra, does its local multiplier algebra—defined as the C*-direct limit of the multiplier algebras of its essential closed ideals—admit only inner derivations? In this talk, we revisit the classical innerness problem for derivations on C*-algebras, highlighting both recent developments and emerging perspectives.