Inverse closed ultradifferential subalgebras (Q2257730)

New methods to generate smooth inverse-closed subalgebras of a given Banach algebra are described and such subalgebras are characterized by approximation properties and weights. Ultradifferentiable classes of Carleman type are introduced and necessary and sufficient conditions for their inverse-closedness are given. Carleman classes satisfying the axiom (M2\(^\prime\)) of \textit{H. Komatsu} [J. Fac. Sci., Univ. Tokyo, Sect. I A 20, 25--105 (1973; Zbl 0258.46039)] are characterized by approximation and weight conditions. Results on the inverse-closedness of Dales-Davie algebras are considered. Some applications to matrix algebras with off-diagonal decay are given.