Composition Operators and Endomorphisms

TL;DR

This paper explores the structure of certain endomorphisms induced by inner functions on spaces of analytic functions, connecting composition operators, Cuntz families, and Hilbert C*-modules.

Contribution

It characterizes endomorphisms of operator algebras implementing composition-induced endomorphisms using advanced operator theory tools.

Findings

Describes the structure of endomorphisms of $B(L^2(\mathbb{T}))$ and $B(H^2(\mathbb{T}))$

Links composition operators with Cuntz families of isometries

Integrates theory of composition operators with Hilbert C*-modules

Abstract

If $b$ is an inner function, then composition with $b$ induces an endomorphism, $\beta$, of $L^\infty(\mathbb{T})$ that leaves $H^\infty(\mathbb{T})$ invariant. We investigate the structure of the endomorphisms of $B(L^2(\mathbb{T}))$ and $B(H^2(\mathbb{T}))$ that implement $\beta$ through the representations of $L^\infty(\mathbb{T})$ and $H^\infty(\mathbb{T})$ in terms of multiplication operators on $L^2(\mathbb{T})$ and $H^2(\mathbb{T})$. Our analysis, which is based on work of R. Rochberg and J. McDonald, will wind its way through the theory of composition operators on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert $C^*$-modules.