On *-representations of a class of algebras with polynomial growth related to Coxeter graphs

Abstract

For a Hilbert space $H$, we study configurations of its subspaces related to Coxeter graphs $\mathbb{G}_{s_1,s_2}$, $s_1,s_2\in\{4,5\}$, which are arbitrary trees such that one edge has type~$s_1$, another one has type~$s_2$ and the rest are of type~$3$. We prove that such irreducible configurations exist only in a finite dimensional $H$, where the dimension of $H$ does not exceed the number of vertices of the graph by more than twice. We give a description of all irreducible nonequivalent configurations; they are indexed with a continuous parameter. As an example, we study irreducible configurations related to a graph that consists of three vertices and two edges of type $s_1$ and $s_2$.