TITLE:
Groupoid Approach to Ergodic Dynamical System of Commutative von Neumann Algebra
AUTHORS:
Nicholas O. Okeke, Murphy E. Egwe
KEYWORDS:
Measure Groupoid, Groupoid Equivalence, Ergodic Action, Convolution Algebra, von Neumann Algebra, Generalized Space
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.14 No.3,
March
26,
2024
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=L2(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) .