# Theory of $B(X)$-module -Algebraic module structure of generally-unbounded infinitesimal generators-

## TL;DR

This paper introduces a logarithmic representation for infinitesimal generators to analyze their algebraic structure, enabling the extraction of bounded parts from unbounded generators and applying this to the rotation group in physics.

## Contribution

It proposes a novel logarithmic representation for unbounded generators and extends the concept of modules over Banach algebras, with applications to mathematical physics.

## Key findings

- Logarithmic representation clarifies algebraic structure of generators
- Bounded components extracted from unbounded generators
- Rigorous formulation of rotation group using Banach algebra modules

## Abstract

The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally-unbounded infinitesimal generators. In conclusion the concept of module over a Banach algebra is proposed as the generalization of Banach algebra. As an application to mathematical physics, the rigorous formulation of rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.