The representation theory of numerical semigroups and the ideal structure of Exel's crossed product

We study representations of numerical semigroups Σ 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*(Σ) generated by a universal isometric representation with commuting range projections, and also prove a structure theorem for C*(Σ). We also investigate the ideal structure of Exel's crossed product C₀(T)⋊_α,Lℕ. We give conditions describing precisely when C₀(T)⋊_α,Lℕ is simple. We provide a complete description of the gauge-invariant ideals of C₀(T)⋊_α,Lℕ, and give a condition which ensures that every ideal of C₀(T)⋊_α,Lℕ is gauge invariant. Under the assumption that Τ is second countable, we describe the primitive ideal structure of C₀(T)⋊_α,Lℕ.