Spectral invariant subalgebras of reduced crossed product C∗-algebras

Let G be a discrete group with a proper length function l. In this paper, we prove that the Schwartz space S2l(G,A) is a spectral invariant dense subalgebra of the reduced crossed product Cr∗(G,A) for all commutative C∗-algebra A with a G-action if and only if G has polynomial growth with respect to l.