Regular orthoscalar representations of the extended Dynkin graph $\widetilde{E}_8$ and ∗-algebra associatedwith it

Abstract

We obtain a classification of regular orthoscalar representations of the extended Dynkin graph $\widetilde{E}_8$ with special character. Using this classification, we describe triples of self-adjoint operators A, B, and C such that their spectra are contained in the sets $\{0,1,2,3,4,5\}, \{0,2,4\}$, and $\{0,3\}$, respectively, and the equality $A + B + C = 6I$ is true.