Multiplicative partial isometries, manageability, and C*-algebraic quantum groupoids

Generalizing the notion of a multiplicative unitary (in the sense of Baaj-Skandalis), which plays a fundamental role in the theory of locally compact quantum groups, we develop in this paper the notion of a multiplicative partial isometry. The axioms include the pentagon equation, but more is needed. Under suitable conditions (such as the "manageability"), it is possible to construct from it a pair of C*-algebras having the structure of a C*-algebraic quantum groupoid of separable type. Generalizing the notion of a multiplicative unitary operator, which plays a fundamental role in the theory of locally compact quantum groups, we develop in this paper the notion of a multiplicative partial isometry. The axioms include the pentagon equation, but more is needed. Under the "manageability" condition on a multiplicative partial isometry (modified from the Woronowicz's condition for a multiplicative unitary), it is possible to construct from it a pair of C*-algebras having almost the structure of a C*-algebraic quantum groupoid of separable type.