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Sep 6, 2010 at 21:35 comment added Yemon Choi On the other hand, there has been interest in recent years in certain algebras of operators generated by representations of special classes of monoid (polycyclic monoids; inverse semigroups; monoids arising from dynamics) so perhaps that is what Jim's answer is referring to
Sep 6, 2010 at 21:33 comment added Yemon Choi It may be worth pointing out that in most cases I know of, semigroups of operators are just images of the positive real line. So in that corner of analysis, it's not clear that "representation theory of the real half-line" is that useful a viewpoint.
Aug 30, 2010 at 13:21 history answered Jim Humphreys CC BY-SA 2.5