The Isometric Representation Theory of Numerical Semigroups

We study representations of numerical semigroups ∑ by isometries on Hilbert space with commuting range projections. Our main theorem says that each such representation is unitarily equivalent to the direct sum of a representation by unitaries and a finite number of multiples of particular concrete representations by isometries. We use our main theorem to identify the faithful representations of the C*-algebra C*(∑) generated by a universal isometric representation with commuting range projections, and also prove a structure theorem for C*(∑). This research was supported by the Australian Research Council.