A subalgebra of the Hardy algebra relevant in control theory and its algebraic-analytic properties

TL;DR

This paper characterizes a specific Banach algebra of functions relevant in control theory, establishing its algebraic and topological properties, including the corona theorem and contractibility, to aid in system stabilization analysis.

Contribution

It provides a complete description of the maximal ideal space and key algebraic properties of the algebra A_0+AP_+, advancing the mathematical foundation for control theory applications.

Findings

Complete description of the maximal ideal space M(A_0+AP_+)

Proof of the corona theorem for A_0+AP_+

M(A_0+AP_+) is contractible, making A_0+AP_+ a projective free ring

Abstract

We denote by A_0+AP_+ the Banach algebra of all complex-valued functions f defined in the closed right half plane, such that f is the sum of a holomorphic function vanishing at infinity and a ``causal'' almost periodic function. We give a complete description of the maximum ideal space M(A_0+AP_+) of A_0+AP_+. Using this description, we also establish the following results: (1) The corona theorem for A_0+AP_+. (2) M(A_0+AP_+) is contractible (which implies that A_0+AP_+ is a projective free ring). (3) A_0+AP_+ is not a GCD domain. (4) A_0+AP_+ is not a pre-Bezout domain. (5) A_0+AP_+ is not a coherent ring. The study of the above algebraic-anlaytic properties is motivated by applications in the frequency domain approach to linear control theory, where they play an important role in the stabilization problem.