The Irreducibility of C*–algebras Acting on Hilbert C*–modules

Abstract

Let B be a C*-algebra, E be a Hilbert B module and L(E) be the set of adjointable operators on E. Let A be a non-zero C*-subalgebra of L(E). In this paper, some new kinds of irreducibilities of A acting on E are introduced, which are all the generalizations of those
associated to Hilbert spaces. The difference between these irreducibilities are illustrated by a number of
counterexamples. It is concluded that for full Hilbert B-modules, these irreducibilities are all equivalent if and only if the underlying C*-algebra B is isomorphic to
the C*-algebra of all compact operators on a Hilbert space.