On Homomorphisms of Douglas Algebras

TL;DR

This paper investigates homomorphisms between Douglas algebras and semisimple Banach algebras, utilizing the structure of continuous mappings into maximal ideal spaces, revealing density and trivial homotopy groups.

Contribution

It provides new insights into the structure of homomorphisms between Douglas algebras and semisimple Banach algebras, especially regarding the topology of continuous mappings and homotopy groups.

Findings

Density of continuous mappings from Z to D in pointwise topology

Trivial homotopy groups of the maximal ideal space

Structural results on homomorphisms between Douglas algebras

Abstract

The paper describes homomorphisms between Douglas algebras and some semisimple Banach algebras. The main tool is a result on the structure of the space $C(Z,\mathfrak M)$ of continuous mappings from a connected first-countable $T_1$ space $Z$ to the maximal ideal space $\mathfrak M$ of the algebra $H^\infty$ of bounded holomorphic functions on the unit disk $\mathbb D\subset\mathbb C$. In particular, it is shown that the space of continuous mappings from $Z$ to $\mathbb D$ is dense in the topology of pointwise convergence in $C(Z,\mathfrak M)$ and the homotopy groups of $\mathfrak M$ are trivial.