Algebras of Almost Periodic Functions with Bohr-Fourier Spectrum in a Semigroup: Hermite Property and its Applications

It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener--Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables.