Properness conditions for actions and coactions

Abstract

Three properness conditions for actions of locally compact groups on C⇤-algebras are studied, as well as their dual analogues for coactions. To motivate the properness conditions for actions, the commutative cases (actions on spaces) are surveyed; here the conditions are known: proper, locally proper, and pointwise properness, although the latter property has not been so well studied in the literature. The basic theory of these properness conditions is summarized, with somewhat more attention paid to pointwise properness. C⇤-characterizations of the properties are proved, and applications to C⇤- dynamical systems are examined. This paper is partially expository, but some of the results are believed to be new.