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.