Noncommutative Approximation: Inverse-closed Subalgebras and Off-diagonal Decay of Matrices

We investigate two systematic constructions of inverse-closed subalgebras of a given Banach algebra or operator algebra A, both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on A and yields a family of smooth inverse-closed subalgebras of A that resemble the usual Hölder-Zygmund spaces. The second construction starts with a graded sequence of subspaces of A and yields a class of inverse-closed subalgebras that resemble the classical approximation spaces. We prove a theorem of Jackson-Bernstein type to show that in certain cases both constructions are equivalent. These results about abstract Banach algebras are applied to algebras of infinite matrices with off-diagonal decay. In particular, we obtain new and unexpected conditions of off-diagonal decay that are preserved under matrix inversion.