With every Coxeter system one can associate a family of algebras considered as deformation of its group algebra. These so-called Hecke algebras, are classical objects of study in combinatorics and representation theory. Complex Hecke algebras admit a natural *-structure and a natural *-representation on a Hilbert space. Taking the norm- and SOT-closure in such representation, one obtains Hecke operator algebras, which have recently seen increased attention. In this talk, I will motivate and introduce Hecke operator algebras, focusing on the case of right-angled Coxeter systems. This case is is particularly interesting from an operator algebraic perspective, thanks to its description by iterated amalgamated free products. I will survey known results on the structure of Hecke operator algebras, before I describe recent joint work with Adam Skalski on the factor decomposition of right-angled Hecke von Neumann algebras as well as the K-theory of right-angled Hecke C*-algebras. If time permits, I will describe some applications to representation theory.
