Abstract
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.
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