A subalgebra of the Hardy algebra relevant in control theory and its algebraic-analytic properties
Abstract
We denote by A0+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(A0+AP+) of A0+AP+. Using this description, we also establish the following results: (1) The corona theorem for A0+AP+. (2) M(A0+AP+) is contractible (which implies that A0+AP+ is a projective free ring). (3) A0+AP+ is not a GCD domain. (4) A0+AP+ is not a pre-Bezout domain. (5) A0+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.
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