Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra

Abstract

Given a compact and regular Hausdorff measure space ( X , μ ), with μ a Radon measure, it is known that the generalised space M ( X ) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L ∞ ( X , μ ) on X . We confirm that the commutative von Neumann algebras M ⊂ B (H), with H= L 2 ( X , μ ), are unitary equivariant to the maximal ideals of the commutative algebra C ( X ). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C ( X ) following its action on M ( X ) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B (H) .