Remarks on duality and semigroups

Duality and algebraically irreducible semigroups

“…The range semigroup in [1] and [2] is the complex unit disk with complex multiplication, while the range in [6] is quite different. In [4], [10], [11] and [12], the homomorphisms were taken to be measurable (with respect to a fixed "nicely" chosen measure) on the given topological semigroup and the range semigroup the complex unit disk. As is well known a measurable homomorphism of a locally compact topological group to the circle group is of necessity continuous.…”

“…As is well known a measurable homomorphism of a locally compact topological group to the circle group is of necessity continuous. Thus it seems to us, especially in view of the results of [4], [11] and [12] that the approach of measurable homomorphisms has an advantage.…”