  In this paper, we study the notion of a separability idempotent in the
C*-algebra framework. This is analogous to the notion in the purely algebraic
setting, typically considered in the case of (finite-dimensional) algebras with
identity, then later also considered in the multiplier algebra framework by the
second-named author. The current work was motivated by the appearance of such
objects in the authors' ongoing work on locally compact quantum groupoids.
