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 $\ensuremath {\mathcal {A}}$, both of which are inspired by classical principles of approximation theory. The first construction requires a closed derivation or a commutative automorphism group on $\ensuremath {\mathcal {A}}$ and yields a family of smooth inverse-closed subalgebras of $\ensuremath {\mathcal {A}}$ that resemble the usual Hölder–Zygmund spaces. The second construction starts with a graded sequence of subspaces of $\ensuremath{\mathcal{A}}$ 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.