Is Every Product System Concrete?

Abstract

Is every product system of Hilbert spaces over a semigroup $P$ concrete, i.e. isomorphic to the product system of an $E_0$-semigroup over $P$? The answer, in general, is no. We record a non-example when $P$ is cancellative and is not embeddable in a group. However, we show that the answer is yes for a reasonable class of semigroups which includes solid, Borel subsemigroups of locally compact abelian groups. We also extend Liebscher's result by showing that in the commutative setting, two product systems are isomorphic if and only if they are algebraically isomorphic.