Finite and Countable Collections of Graded-Commuting Self-Adjoint Operators (GCSO)

Abstract

In this chapter we study both finite and countable collections of self-adjoint operators which are mutually connected by relations of commutation or anticommutation (graded commutation). To describe which operators of the collection commute or anticommute, we can consider the operators of the collection as operators of the self- adjoint representation of an unordered simple graph Γ. There the vertices of the graph Γ correspond to the operators of the collection (A _k ). If there is an edge between the vertices a _i and a _j , it means that the operators A _i and A _j anticommute. If there is no edge, it means that they commute. For finite and “tame” countable collections of graded-commuting operators we can prove a spectral theorem, for “wild” countable collections we study particular classes of representations and the objects which arise: measures and cocycles.