On *-representations of a certain class of algebras related to a graph

Abstract

We study families of self-adjoint operators with given spectra whose sum is a scalar operator. Such families are $\ast$-representations of certain algebras which can be described in terms of graphs and positive functions on them. The main result is that in the cases where the graph is one of the extended Dynkin graphs ${\tilde{D}}_4$, ${\tilde{E}}_6$, ${\tilde{E}}_7$ or ${\tilde{E}}_8$, all irreducible $\ast$-representations of the corresponding algebra are finite-dimensional. To prove this fact, we introduce the notion of invariant functional on a graph and give their description.

7 pages. Submitted to MFAT

References in corpus (2)

• Locally-scalar representations of graphs in the category of Hilbert spaces
• Certain Examples of Deformed Preprojective Algebras and Geometry of Their *-Representations

Cited by in corpus (2)

• Special characters on star graphs and representations of $\ast$-algebras
• $n$-Subspaces in linear and unitary spaces