# Monoids, their boundaries, fractals and $C^\ast$-algebras

## TL;DR

This paper explores the connections between self-similar fractals associated with monoids and the boundary quotients of related $C^\ ext{*}$-algebras, revealing new examples and raising open questions in the field.

## Contribution

It introduces a framework linking monoid fractals to boundary quotients of $C^*$-algebras, extending prior concepts and providing new examples.

## Key findings

- Existence of self-similar M-fractals leads to new $C^*$-algebra examples.
- The boundary of a finitely 1-generated monoid has two natural topologies.
- The cone topology allows defining measures on fractal attractors.

## Abstract

In this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]) and the theory of boundary quotients of $C^\ast$-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar M-fractals for a given monoid M, gives rise to examples of $C^\ast$- algebras generalizing the boundary quotients discussed by X. Li in [4, {\S}7, p. 71]. The starting point for our investigations is the observation that the universal boundary of a finitely 1-generated monoid carries naturally two topologies. The fine topology plays a prominent role in the construction of these boundary quotients. On the other hand, the cone topology can be used to define canonical measures on the attractor of an M-fractal provided M is finitely 1-generated.