# Effective descent morphisms for Banach modules

## TL;DR

This paper characterizes effective descent morphisms in the category of Banach modules as exactly those norm-decreasing algebra homomorphisms that are weak retracts, providing a precise criterion.

## Contribution

It establishes a necessary and sufficient condition for effective descent morphisms in Banach modules, linking algebraic and categorical properties.

## Key findings

- Effective descent morphisms are characterized as weak retracts.
- Norm-decreasing homomorphisms are effective descent morphisms iff they are weak retracts.
- Provides a categorical criterion for Banach algebra homomorphisms.

## Abstract

It is proved that a norm-decreasing homomorphism of commutative Banach algebras is an effective descent morphism for Banach modules if and only if it is a weak retract.