On \(*\)-representations of a class of algebras with polynomial growth related to Coxeter graphs (Q2896646)

Configurations of subspaces of a Hilbert space or, equivalently, \(*\)-algebras generated by projections which correspond to Coxeter graphs were studied by \textit{N. D. Popova}, \textit{Yu. S. Samojlenko} and \textit{A. V. Strelets} [Ukr. Math. J. 59, No. 6, 907--918 (2007; Zbl 1150.16022); ibid. 60, No. 4, 623--638 (2008; Zbl 1164.16005); Operator Theory: Advances and Applications 190, 429--450 (2009; Zbl 1184.46026)]. Their properties depend on the geometric characteristics of a graph.NEWLINENEWLINEIn the paper under review, the authors consider the case of the Coxeter graphs \(\mathbb G_{s_1,s_2}\), \(s_1,s_2 = {4,5}\), which are arbitrary trees such that one edge has type \(s_1\), another has type \(s_2\), while the rest are of type 3. It is proved that such irreducible configurations exist only in a finite-dimensional space \(\mathcal H\) with \(\dim \mathcal H\leq 2n\) (\(n\) is the number of vertices). A description of all non-equivalent irreducible configurations is given.