Algebras of reflections in $$L^2(\mathbb {T})$$

Let $$\mathbb {D}=\{z\in \mathbb {C}: |z|<1\}$$ and $$\mathbb {T}=\{z\in \mathbb {C}: |z|=1\}$$ . For $$a\in \mathbb {D}$$ , consider $$\varphi _a(z)=\displaystyle {\frac{a-z}{1-\bar{a}z}}$$ and $$C_a$$ the composition operator in $$L^2(\mathbb {T})$$ induced by $$\varphi _a$$ : $$\begin{aligned} C_a f=f\circ \varphi _a. \end{aligned}$$ Clearly $$C_a$$ satisfies $$C_a^2=I$$ , i.e., is a non-selfadjoint reflection. In this paper we study the operator algebras related to $$C_a$$ : the $$\text {C}^*$$ -algebra generated by $$C_a$$ , its commutant and its double commutant.