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

Abstract

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

Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/