# Journal of Operator Theory

Volume 67, Issue 2, Spring 2012 pp. 581-604.

A semigroup composition $C^*$-algebra**Authors**: Katie S. Quertermous

**Author institution:**Department of Mathematics and Statistics, James Madison University, Harrisonburg, VA 22807, U.S.A.

**Summary:**For $0 < s < 1,$ let $\varphi_s(z)=sz+(1-s).$ We investigate the unital $C^*$-algebra generated by the semigroup $\{C_{\varphi_s} : 0 < s < 1\}$ of composition operators acting on the Hardy space of the unit disk. We determine the joint approximate point spectrum of a related collection of operators and show that the quotient of the $C^*$-algebra by its commutator ideal is isomorphic to the direct sum of $\IC$ and the algebra of almost periodic functions on the real line. In addition, we show that the $C^*$-algebra is irreducible.

**Keywords:**composition operator, Toeplitz operator, Hardy space, almost periodic function, $C^*$-algebra, commutator ideal

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