On the Existence of Configurations of Subspaces in a Hilbert Space with Fixed Angles

Abstract

For a class of $*$-algebras, where $*$-algebra $A_{Γ,τ}$ is generated by projections associated with vertices of graph $Γ$ and depends on a parameter $τ$ $(0 < τ\leq 1)$, we study the sets $Σ_Γ$ of values of $τ$ such that the algebras $A_{Γ,τ}$ have nontrivial $*$-representations, by using the theory of spectra of graphs. In other words, we study such values of $τ$ that the corresponding configurations of subspaces in a Hilbert space exist.