The Local Multiplier Algebra of a C*-Algebra, II

Abstract

Let A be a separable C*-algebra and let M_loc(A) be the local multiplier algebra of A. It is shown that every minimal closed prime ideal of M_loc(A) is primitive. If Prim(A) has a dense G_δ consisting of closed points (for instance, if Prim(A) is a T₁-space) and A is unital, then M_loc(A) is its own local multiplier algebra and has only inner derivations. The same is true for M_loc(M_loc(A)) if A is non-unital. If A is postliminal then M_loc(M_loc((A)) is the regular σ-completion of A, which is an AW*-algebra.