Elementary operators on the algebra of adjointable operators on a Hilbert module

Abstract

Let A be a prime unital C⁎-algebra, X a countably generated Hillbert A-module, B(X) the C⁎-algebra of adjointable operators on X and K(X) the C⁎-algebra of (generalised) compact operators on X. We characterise multiplication operators and elementary operators on B(X) in terms of the size of their images. To obtain these characterisations we introduce the concept of a uniformly approximable subset of a C⁎-algebra. We show that MA,B(B(X))⊆K(X) if and only if at least one of A or B belongs to K(X). We show that the set MA,B(B(X)1) is a uniformly approximable subset of K(X), (B(X)1 is the unit ball of B(X)), if and only if A,B∈K(X). If Φ is an elementary operator on B(X), we show that Φ(B(X))⊆K(X) (resp. is a uniformly approximable subset of K(X)) if and only if there exist {Ai}i=1k,{Bi}i=1k⊆B(X) such that at least one of Ai or Bi (resp. both) belong to K(X) for i=1,…,k and Φ=∑i=1kMAi,Bi.