The C$^*$-algebra of a composition reflection
Esteban Andruchow
TL;DR
This work analyzes the C$^*$-algebra generated by the composition operator $C_a$ on the Hardy space $H^2$, where $C_af=f\circ\varphi_a$ and $\varphi_a$ is the disk automorphism $\varphi_a(z)=\frac{a-z}{1-\bar{a}z}$. It shows ${\cal C}^*(C_a)$ is generated by the two eigenspace projections $P_{N(C_a-I)}$ and $P_{N(C_a+I)}$, and derives detailed spectral information for the associated projection product $\mathbf{\Delta}_a$, yielding $\|\mathbf{\Delta}_a\|=|a|^2$ and $\sigma(\mathbf{\Delta}_a)=[0,|a|^2]$. The paper then places ${\cal C}^*(C_a)$ within Pedersen’s framework, adapting it via Spitkovsky’s results to obtain a concrete, parameterized description that shows ${\cal C}^*(C_a)\cong {\cal C}^*(C_b)$ for any $a,b\neq 0$, i.e., the algebra depends only on $|a|$. Connections with the boundary-model operator $\Gamma_a$, two symmetries $\rho_a$ and $W_a$, and a Toeplitz intertwinor $T_{\varphi_{\omega_a}}$ are developed, including an explicit intertwining relation $T_{\varphi_{\omega_a}}^*C_aT_{\varphi_{\omega_a}}=-C_a$ and the fact that $W_{\omega_a}C_aW_{\omega_a}=T_{f_a}C_0$. Together, these results illuminate the internal symmetries and spectral geometry governing the composition-operator C$^*$-algebras on the disk.
Abstract
We study the C$^*$ algebra generated by the composition operator $C_a$ acting on the Hardy space $H^2$ of the unit disk, given by $C_af=f\circ\varphi_a$, where $$ \varphi_a(z)=\frac{a-z}{1-\bar{a}z}, $$ for $|a|<1$. Also several operators related to $C_a$ are examined.