Smooth subalgebras of the Roe algebra

In this paper, for a discrete group with property (RD), we construct a smooth subalgebra of a certain subalgebra in the (uniform) Roe algebra of this group. And using Lafforgue’s [Formula: see text]-Theory, under certain conditions, we prove that this certain subalgebra and the (uniform) Roe algebra have the same [Formula: see text]-theory groups. Moreover, our smooth subalgebra and the (uniform) Roe algebra have the same [Formula: see text]-theory groups. Our result can be viewed as a smooth subalgebra construction of the (uniform) Roe algebra, which is, as far as we know, the first occurrence in literature.

CHARACTERIZATIONS OF LOCALLY FINITE ACTIONS OF GROUPS ON SETS

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.