Inverse-closedness of subalgebras of integral operators with almost periodic kernels

The integral operator of the form $$\left(Nu\right)(x)=\sum _{k=1}^{\mathrm{\infty }}{e}^{i⟨{\omega }_k,x⟩}{\int }_{{\mathbb{R}}^c}{n}_k(x-y)u(y)dy$$ acting in $L_p({\mathbb{R}}^c)$, $1\le p\le \mathrm{\infty }$, is considered. It is assumed that ${\omega }_k\in {\mathbb{R}}^c$, $n_k\in L_1({\mathbb{R}}^c)$, and $$\sum _{k=1}^{\mathrm{\infty }}\parallel n_k{\parallel }_{L_1}<\mathrm{\infty }\mathrm{.}$$ We prove that if the operator $\mathbf{1}+N$ is invertible, then $(\mathbf{1}+N{)}^{-1}=\mathbf{1}+M$, where $M$ is an integral operator possessing the analogous representation.