The C${}^{\ast }$-algebra of a composition reflection

We study the C${}^{\ast }$ algebra generated by the composition operator $C_a$ acting on the Hardy space $H^2$ of the unit disk, given by $C_{a}f=f\circ {\phi }_a$, where $${\phi }_a(z)=\frac{a-z}{1-\bar{a}z},$$ for $\mathrm{\mid }a\mathrm{\mid }<1$. Also several operators related to $C_a$ are examined.