Title: A class of C*-algebraic locally compact quantum groupoids Part I. Motivation and definition,
Abstract: In this series of papers, we develop the theory of a class of locally compact
quantum groupoids, which is motivated by the purely algebraic notion of weak
multiplier Hopf algebras. In this Part I, we provide motivation and formulate
the definition in the C*-algebra framework. Existence of a certain canonical
idempotent element is required and it plays a fundamental role, including the
establishment of the coassociativity of the comultiplication. This class
contains locally compact quantum groups as a subclass.