The second local multiplier algebra of a separable C*-algebra

TL;DR

This paper surveys the construction of separable C*-algebras with a strictly larger second local multiplier algebra than the first, and introduces criteria for when these algebras have equal first and second local multiplier algebras.

Contribution

It provides a systematic approach to produce examples with larger second local multiplier algebras and new criteria for their equality with the first, including conditions for inner derivations.

Findings

Examples of C*-algebras with larger second local multiplier algebra identified.

Criteria established for when the first and second local multiplier algebras coincide.

All derivations of the local multiplier algebra are inner for certain classes.

Abstract

Several examples of (separable) C*-algebras with the property that their second (iterated) local multiplier algebra is strictly larger than the first have been found by various groups of authors over the past few years, thus answering a question originally posed by G. K. Pedersen in 1978. This survey discusses a systematic approach by P. Ara and the author to produce such examples on the one hand; on the other hand, we present new criteria guaranteeing that the second and the first local multiplier algebra of a separable C*-algebra agree. For this class of C*-algebras, each derivation of the local multiplier algebra is inner.