Journal of Operator Theory

Volume 62, Issue 2, Fall 2009 pp. 357-370.

The isometric representation theory of a perforated semigroup

Authors: 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 $\Sigma:=\N \setminus \{ 1 \}$ of $\N$, and study representations of $\Sigma$ 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 $\ell^{2} (\Sigma)$, and a multiple of a representation by shifts on $\ell^{2} ( \N )$. We consider also the $C^*$-algebra $C^*(\Sigma)$ generated by a universal isometric representation with commuting range projections, and use our main theorem to identify the faithful representations of $C^*(\Sigma)$ and prove a structure theorem for $C^*(\Sigma)$.

Keywords: Perforated semigroup, isometric representation, Wold decomposition

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