The type III manufactory

TL;DR

This paper explores the construction of type III factors in von Neumann algebra theory using unconventional objects like the game of Go and Conway's Game of Life, generalized to finitely presented groups.

Contribution

It introduces a novel method of building type III factors through the use of unusual objects and games, expanding the toolkit for von Neumann algebra construction.

Findings

Constructed many type III factors using game-theoretic objects.

Demonstrated the applicability of generalized Conway games in algebraic structures.

Provided new insights into the interplay between game theory and operator algebras.

Abstract

Using unusual objects in the theory of von Neumann algebra, as the chinese game Go or the Conway game of life (generalized on finitely presented groups), we are able to build, by hands, many type III factors.