We discuss proper outerness for finite-index endomorphisms and finite-index bimodules of simple C \(^*\) -algebras. The notion of properly outer automorphisms of von Neumann algebras was introduced by Connes, and the C \(^*\) -algebraic analogue for automorphisms has also been extensively investigated. Proper outerness is a strengthened form of outerness for automorphisms, endomorphisms, and bimodules of C \(^*\) -algebras, and may be viewed as a noncommutative analogue of topological freeness for homeomorphisms of locally compact spaces. It is a classical theorem of Kishimoto that every outer automorphism of a simple C \(^*\) -algebra is properly outer. As an application, he proved the simplicity of crossed product C \(^*\) -algebras arising from outer actions of discrete groups on simple C \(^*\) -algebras. More recently, Izumi proved that finite-index outer endomorphisms of purely infinite simple C \(^*\) -algebras are automatically properly outer. In this article, we extend Izumi’s result beyond the purely infinite setting. Our main result is that every finite-index outer endomorphism of a simple C \(^*\) -algebra is properly outer. Understanding finite-index endomorphisms is useful for studying symmetries beyond group actions, including actions by unitary tensor categories. As a consequence of our main result, freeness for outer actions of unitary tensor categories on simple C \(^*\) -algebras is also shown to hold automatically. As applications, we obtain structural results about irreducible discrete inclusions of C \(^*\) -algebras, potentially having infinite index, such as C \(^*\) -irreducibility in the sense of Rørdam.
