A semigroup composition \(C^*\)-algebra (Q2891161)

Several authors have recently studied unital \(C^*\)-algebras generated by composition operators. Most of these investigations focus on composition operators induced by linear fractional self maps on the unit disc \(\mathbb D\) of the complex plane. These investigations split in two cases. The automorphism case was investigated by \textit{M. T. Jury} [Indiana Univ. Math. J. 56, No.~6, 3171--3192 (2007; Zbl 1153.46041)] and the non-automorphism case by \textit{T. L. Kriete} et al.\ [J. Oper. Theory 58, No.~1, 135--156 (2007; Zbl 1134.47303)].NEWLINENEWLINEThe paper under review begins the consideration of the remaining non-automorphism case. More precisely, for \(0<s<1\), set \(\varphi_s(z):=sz+(1-s)\), \(z\in \mathbb D\). The unital \(C^*\)-algebra generated by the semigroup \(\{C_{\varphi_s} : \;0<s<1 \}\) of composition operators acting on the Hardy space \(H^2\) on the disc is investigated. The author determines the joint approximate point spectrum of a related collection of operators, and shows in the main result that the quotient of the \(C^*\)-algebra by its commutator ideal is isomorphic to the direct sum of \(\mathbb C\) and the algebra of almost periodic functions on the real line. In the last section, it is proved that the \(C^*\)-algebra under consideration is irreducible.