Operator algebras and groups acting on right-angled buildings

We propose to pursue in-depth study of symmetry groups of certain regular discrete structures by modern operator algebraic methods.

Specifically, automorphism groups of right-angled buildings, thought of as regularly assembled from hypercubes of varying dimension, will be studied.

Their unitary representation theory, which describes quantum systems with prescribed symmetries, is not accessible by classical methods.

However, methods from operator algebras including noncommutative geometry, boundary theory and free probability theory can shed new light on these groups.

Particular attention is given to right-angled Hecke operator algebras, which encode the most complicated part of representation theory.We plan to complete three different projects, concerned with A) the division of representations into tame and wild ones B) noncommutative geometry and classification of right-angled Hecke C*-algebras and C) fine structure of right-angled Hecke operator algebras.

Each project is subdivided into intermediate goals, which are the basis of a time plan and guarantee feasibility.A quickly developing theory of symmetry groups of discrete structures creates a need to also understand their representation theory. The class of groups we propose to study is rich enough to derive a blueprint for future research.

Yet it is particularly well tractable by operator algebraic methods, thanks to structural similarity with free product constructions, which have been studied for a long time.