We study how paradoxicality properties affect the way groups partially acton topological spaces and C*-algebras. We also investigate the real rank zero and AF properties for certain classes of group C*-algebras.
Specifically, in article A, we characterize supramenable groups in terms of existence of invariant probability measures for partial actions on compact Hausdorff spaces and existence of tracial states on partial crossed products. These characterizations show that, in general, one cannot decompose a partial crossed product of a C*-algebra by a semidirect product of groups as two iterated partialcrossed products. We give conditions which ensure that such decomposition is possible.
In Article B, we show that an action of a group on a set X is locally finite if and only if X is not equidecomposable with a proper subset of itself. As a consequence, a group is locally finite if and only if its uniform Roe algebra is finite.
In Article C, we analyze the C*-algebra generated by the Koopman representation of a topological full group, showing, in particular, that it is not AF andhas real rank zero. We also prove that if G is a finitely generated, elementary amenable group, and C*(G) has real rank zero, then G is finite.