The Centre of the Local Multiplier Algebra

Abstract

In this chapter, we shall investigate C*-algebras A whose centre is ‘rich’ in comparison with the ideal structure of A. If A is a simple unital C*-algebra, the one-dimensional centre Z(A) contains sufficient information. But if A is merely prime, a one-dimensional centre appears to be ‘small’ compared with a possibly large lattice of closed ideals. Yet, in this case, M _loc (A) too is prime, whence dim Z(M _loc (A)) = 1 (Proposition 3.3.2). As it emerges, it is the centre Z(M _loc (A)) of the local multiplier algebra of A rather than Z(A), or Z(M(A)) in the non-unital case, that contains information about properties of operators defined on A which are compatible with ideals of A (such as derivations, automorphisms, elementary operators, etc.). This will be the theme of the subsequent chapters, where it will become evident why the behaviour of these classes of operators on prime C*-algebras is so neat; e.g., th e norm of an inner derivation °_a on a prime C*-algebra A simply equals twice the distance from a to Z(M(A)) (Corollary 4.1.21).