The inverse-closed subalgebra of $C^{\ast }(G,A)$

This paper studies the inverse-closed subalgebras of the Roe algebra with coefficients of the type $l^2(G,A)$. The coefficient $A$ is chosen to be a non-commutative $C^{\ast }$-algebra, and the object of study is $C^{\ast }(G,A)$ generated by the countable discrete group $G$. By referring to the Sobolev-type algebra, the intersection of a family of Banach algebras is taken. It is proved that the intersection $W_a^{\mathrm{\infty }}(G,A)$ of Banach spaces is a spectrally invariant dense subalgebra of $C^{\ast }(G,A)$, and a sufficient condition for this is that the group action of $G$ has polynomial growth.