Truncation Approximations and Spectral Invariant Subalgebras in Uniform Roe Algebras of Discrete Groups

In this paper we study band truncation approximations for operators in uniform Roe algebras of countable discrete groups. Under conditions on certain growth rates for discrete groups, we find large classes of dense subspaces of uniform Roe algebras whose elements can be approximated by their band truncations in the operator norm. We apply these results to construct a nested family of spectral invariant Banach algebras on discrete groups. For a group with polynomial growth, the intersection of these Banach algebras is a spectral invariant dense subalgebra of the uniform Roe algebra. For a group with subexponential growth, we show that the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra.