We study the algebras of bounded holomorphic functions on the unit disk whose boundary values, having, in a sense, the weakest possible discontinuities, belong to the algebra of semi-almost periodic functions on the unit circle. The latter algebra contains as a special case an algebra introduced by Sarason in connection with some problems in the theory of Toeplitz operators. We show that such algebras have the Grothendieck approximation property, prove the corona theorem for them and formulate some results on the structure of their maximal ideal spaces. Also, we extend the notion of the Bohr–Fourier spectrum to holomorphic semi-almost periodic functions and prove that under certain assumptions on their spectra the corresponding algebras are projective free and their maximal ideal spaces have trivial Cech cohomology groups.
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