A Subalgebra of the Hardy Algebra Relevant in Control Theory and Its Algebraic-Analytic Properties

Abstract

We denote by \(A_{0} + AP_{+}\) the Banach algebra of all complex-valued functions f defined in the closed right halfplane, 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 \(\mathfrak{M}(A_{0} + AP_{+})\) of \(A_{0} + AP_{+}\). Using this description, we also establish the following results: 1. The corona theorem for A 0 + AP +. 2. \(\mathfrak{M}(A_{0} + AP_{+})\) 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.KeywordsMaximal ideal spaceCorona theoremContractabilityGCDPre-BezoutCoherenceSubject ClassificationsPrimary 30H80Secondary 46J2093D1530H05