Discrete Inclusions of C*-algebras

Abstract

We introduce the category of C*-discrete inclusions of C*-algebras $A\subset B$ with a faithful conditional expectation $E:B\twoheadrightarrow A$. This class includes many examples such as finite Watatani index inclusions, and also abundant infinite index inclusions like crossed products by outer actions of discrete (quantum) groups and unitary tensor categories. We prove irreducible $(A'\cap B= \mathbb{C}1)$ C*-discrete inclusions are precisely crossed products by outer actions of unitary tensor categories and certain C*-algebra objects.