Monoids, their boundaries, fractals and C*-algebras

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*-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*-algebras (1.9) generalizing the boundary quotients \partial C_\lambda ^*(\mathcal{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 ℳ-fractal for a finitely 1-generated monoid ℳ.

Case Report: Consecutive Adrenal Cushing’s Syndrome and Cushing’s Disease in a Patient With Somatic CTNNB1, USP8, and NR3C1 Mutations

“…Moreover, exome sequencing excluded a causative germline mutation of PRKAR1A but also of MEN1. With respect to the latter, Dal Verme et al (11) reported the case of a patient who suffered from ectopic Cushing's syndrome due to a thymic carcinoid 32 years after being treated for Cushing's disease, possibly pointing to a MEN1 germline mutation.…”