Introverted algebras with mean value

TL;DR

This paper explores the structure of introverted algebras with mean value, establishing their spectrum as a compact topological semigroup and linking the mean value to Haar integrals, with applications to nonlocal PDE homogenization.

Contribution

It characterizes the spectrum of introverted algebras with mean value as a compact topological semigroup and introduces convolution on this spectrum, connecting harmonic analysis with homogenization theory.

Findings

Spectrum of A is a compact topological semigroup.

Kernel of the spectrum is a compact topological group.

Convolution over the spectrum aids in studying nonlocal PDE asymptotics.

Abstract

Let A be an introverted algebra with mean value. We prove that its spectrum \Delta (A) is a compact topological semigroup, and that the kernel K(\Delta (A)) of \Delta (A) is a compact topological group over which the mean value on A can be identified as the Haar integral. Based on these facts and also on the fact that K(\Delta (A)) is an ideal of \Delta (A), we define the convolution over \Delta (A). We then use it to derive some new convergence results involving the convolution product of sequences. These convergence results provide us with an efficient method for studying the asymptotics of nonlocal problems. The obtained results systematically establish the connection between the abstract harmonic analysis and the homogenization theory. To illustrate this, we work out some homogenization problems in connection with nonlocal partial differential equations.