# On the relative bicentralizer flows and the relative flow of weights of inclusions of factors of type III$_1$

## TL;DR

This paper demonstrates the isomorphism between the relative bicentralizer flow and the relative flow of weights for certain inclusions of type III$_1$ factors, expanding understanding of their structural relationships.

## Contribution

It establishes the isomorphism between the relative bicentralizer flow and the relative flow of weights for specific classes of type III$_1$ factor inclusions.

## Key findings

- Proves the isomorphism for inclusions of injective type III$_1$ factors with finite index.
- Shows the isomorphism for irreducible discrete inclusions with small algebra as an injective type III$_1$ factor.
- Enhances the understanding of the structure of type III$_1$ factor inclusions.

## Abstract

We show the relative bicentralizer flow and the relative flow of weights are isomorphic for an inclusion of injective factors of type III$_1$ with finite index, or an irreducible discrete inclusion whose small algebra is an injective factor of type III$_1$.