Dynamical Maps and Symmetroids

Abstract

Starting from the canonical symmetroid [Formula: see text] associated with a groupoid [Formula: see text], the issue of describing dynamical maps in the groupoidal approach to quantum mechanics is addressed. After inducing a Haar measure on the canonical symmetroid [Formula: see text], the associated von Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid [Formula: see text] and given subsets of functions are associated with completely positive maps. Some simple examples are also presented.