  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.
