# A von Neumann algebraic approach to self-similar group actions

## TL;DR

This paper explores the connection between self-similar group actions and operator algebras, demonstrating that certain von Neumann algebras derived from these actions are type III factors, with KMS states characterized by Bernoulli measures.

## Contribution

It introduces a von Neumann algebraic framework for analyzing self-similar group actions and characterizes the resulting von Neumann algebras as type III factors.

## Key findings

- KMS states correspond to Bernoulli measures
- Von Neumann algebras are type III factors
- Establishes a von Neumann algebraic perspective on self-similar actions

## Abstract

We study some relations between self-similar group actions and operator algebras. We consider KMS states on the Cuntz--Pimsner algebras constructed by Nekrashevych from self-similar actions and the GNS representations of the KMS states. The KMS states are given by the Bernoulli measure. We also consider the von Neumann algebras on the GNS spaces and show that the von Neumann algebras are type III factors.