Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group

In the paper Wiener-Hopf operators on a semigroup of nonnegative elements of a linearly quasi-ordered torsion free Abelian group are considered. Wiener-Hopf factorization of an invertible element of the group algebra is constructed, notions of a topological index and a factor index are introduced. It turns out that the set of factor indices for invertible elements of the group algebra is a linearly ordered group. It is shown that Wiener-Hopf operator with an invertible symbol is an one-side invertible operator and its invertibility properties are defined by the sign of the factor index of its symbol. Groups on which there exist nontrivial Fredholm Wiener-Hopf operators are described. As an example, all linear quasi-orders on the group ⁿ are found and corresponding Wiener-Hopf operators are considered.