On the algebra of integral operators with involution

In this paper, we consider integral operators with kernels depending on the sum and difference of arguments in the space $L_p(\mathbb{R})$, $p\in[1, \infty)$. We prove that such operators form a subalgebra of the algebra of bounded linear operators. The study of operators with kernels depending on the difference of arguments was carried out using Banach $L_1(\mathbb{Z})$-modules. The differences and similarities between the subalgebra of integral operators and the corresponding subalgebra of difference operators with involution are noted.