Monoids, their boundaries, fractals, and C*-algebras

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∗-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar-fractals for a given monoid, gives rise to examples of C∗-algebras (1.9) generalizing the boundary quotients Cλ∗ Cλ ∗(M) discussed by X. Li in [4, §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, while the cone topology can be used to define canonical measures on the attractor of an-fractal for a finitely 1-generated monoid.