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.