On *-representations of algebras related to Dynkin graphs and extended Dynkin graphs

Abstract:
A family of self-adjoint operators with given spectra whose sum is a multiple of the identity is regarded as a representation of the corresponding *-algebra, which can be defined in terms of a graph together with a family of numbers associated to its vertices.

We study conditions under which such families of operators exist and present some facts about the structure of such families for the cases where the graph is Dynkin graph of extended Dynkin graph.