Irreducible inclusions of simple C$^*$-algebras

The literature contains interesting examples of inclusions of simple C$^*$-algebras with the property that all intermediate C$^*$-algebras likewise are simple. In this article we take up a systematic study of such inclusions, which we refer to as being C$^*$-irreducible by the analogy that all intermediate von Neumann algebras of an inclusion of factors are again factors precisely when the given inclusion is irreducible. We give an intrinsic characterization of when an inclusion of C$^*$-algebras is C$^*$-irreducible, and use this to revisit known and exhibit new C$^*$-irreducible inclusions arising from groups and dynamical systems. Using a theorem of Popa one can show that an inclusion of II$_1$-factors is C$^*$-irreducible if and only if it is irreducible with finite Jones index. We further show how one can construct C$^*$-irreducible inclusions from inductive limits, and we discuss how the notion of C$^*$-irreducibility behaves under tensor products.