Inverse closedness of approximation algebras

We prove the inverse closedness of certain approximation algebras based on a quasi-Banach algebra X using two general theorems on the inverse closedness of subspaces of quasi-Banach algebras. In the first theorem commutative algebras are considered while the second theorem can be applied to arbitrary X and to subspaces of X which can be obtained by a general K-method of interpolation between X and an inversely closed subspace Y of X having certain properties. As application we present some inversely closed subalgebras of C(T) and C[−1, 1]. In particular, we generalize Wiener’s theorem, i.e., we show that for many subalgebras S of l1(Z), the property {ck(f )} ∈ S (ck(f ) being the Fourier coefficients of f ) implies the same property for 1/f if f ∈ C(T) vanishes nowhere on T.  2005 Elsevier Inc. All rights reserved