Journal of Operator Theory
Volume 62, Issue 2, Fall 2009 pp. 357-370.
The isometric representation theory of a perforated semigroupAuthors: Iain Raeburn (1) and Sean T. Vittadello (2)
Author institution: (1) School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
(2) School of Mathematical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
Summary: We consider the additive subsemigroup Σ:=\N∖{1} of \N, and study representations of 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 unitary representation, a multiple of the Toeplitz representation on ℓ2(Σ), and a multiple of a representation by shifts on ℓ2(\N). We consider also the C∗-algebra C∗(Σ) generated by a universal isometric representation with commuting range projections, and use our main theorem to identify the faithful representations of C∗(Σ) and prove a structure theorem for C∗(Σ).
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