Noncommutative Approximation: Inverse-Closed Subalgebras and Off-Diagonal Decay of Matrices

TL;DR

This paper develops two methods for constructing inverse-closed subalgebras in Banach algebras, inspired by approximation theory, and applies these to matrices with off-diagonal decay, revealing new decay preservation conditions.

Contribution

It introduces two systematic constructions of inverse-closed subalgebras based on approximation principles and proves their equivalence in certain cases, with applications to matrix off-diagonal decay.

Findings

New conditions of off-diagonal decay preserved under inversion

Equivalence of smooth and approximation-based subalgebra constructions

Application to infinite matrices with decay properties

Abstract

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 Holder-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.