The Irreducibility of $C^*$-algebras Acting on Hilbert $C^*$-modules

Runliang Jiang

Let $\mathcal B$ be a $C^*$-algebra, $E$ be a Hilbert $\mathcal B$ module and $\mathbb L(E)$ be the set of adjointable operators on $E$. Let $\mathcal A$ be a non-zero $C^*$-subalgebra of $\mathbb L(E)$. In this paper, some new kinds of irreducibilities of $\mathcal A$ 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 a full Hilbert $\mathcal B$-module, these irreducibilities are all equivalent if and only if the underlying $C^*$-algebra $\mathcal B$ is isomorphic to the $C^*$-algebra of all compact operators on a Hilbert space.