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

Abstract

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 $C_{a}f=f\circ \ddot{I}{†}_a$, where $$\ddot{I}{†}_a(z)=\frac{a-z}{1-\bar{a}z},$$ for $|a|<1$. Also several operators related to $C_a$ are examined.