Is every product system concrete?

Is every product system of Hilbert spaces over a semigroup concrete, that is, isomorphic to the product system of an ‐semigroup over ? The answer is no if is discrete, cancellative and does not embed in a group. However, we show that the answer is yes for a reasonable class of semigroups. More precisely, let be a Borel subsemigroup with non‐empty interior of a locally compact, second countable, Hausdorff topological group such that one of the following conditions hold. The group is discrete and is normal in , that is, for every . The group is abelian. The semigroup is right Ore, that is, is a group, and there exists such that is an order unit for . Then, every product system of Hilbert spaces over is isomorphic to the product system of an ‐semigroup over . We also extend a result of Liebscher. We prove that, in the setting of abelian subsemigroups of locally compact groups, any two measurable structures on a product system differ by a character (possibly non‐measurable), which in turn implies that two product systems are isomorphic if and only if they are algebraically isomorphic.