Partially Ordered Groups, Almost Invariant Sets, and Toeplitz Operators

For a partially ordered group G, we prove that G+ is almost invariant in G if and only if G is a finite extension of Z. For such partially ordered groups the theory of generalized Toeplitz operators reduces to the corresponding theory for almost invariant sets. If, in addition, G is abelian, then the associated operators are essentially finite direct sums of classical Toeplitz operators. If G is an ordered group, then G+ is almost invariant in G if and only if G is order isomorphic to Z. Thus, for ordered groups, the two theories of generalized Toeplitz operators intersect only in the classical theory.