Given a compact and regular Hausdorff measure space (X, μ), with μa 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) .
References
[1]
Okeke, O.N. and Egwe, M.E. (2020) Cohomogeneity One Groupoid Analysis of the Dynamical System of Rings of Continuous Functions. arXiv: 2101.08159.
[2]
Stratila, S.V. and Zsido, L. (2019) Lectures on von Neumann Algebras. 2nd Edition, Cambridge University Press, Cambridge. https://doi.org/10.1017/9781108654975
[3]
Connes, A. (1994) Noncommutative Geometry. Academic Press, San Diego.
[4]
Berberian, S.K. (2009) Notes on Spectral Theory. 2nd Edition. D. Van Nostrand Company Inc., Princeton.
[5]
Palais, R.S. (1961) On the Existence of Slices for Actions of Non-Compact Lie Groups. Annals of Mathematics, 73, 295-323. https://doi.org/10.2307/1970335
[6]
Palais, R.S. and Terng, C. (1987) A General Theory of Canonical Forms. Transactions of the American Mathematical Society, 300, 771-789. https://doi.org/10.2307/2000369
[7]
Podesta, F. (1995) Cohomogeneity One Riemannian Manifolds and Killing. Differential Geometry and Its Applications, 5, 311-320. https://doi.org/10.1016/0926-2245(95)00021-6
[8]
Einsiedler, M. and Ward, T. (2011) Ergodic Theory. Springer Verlag, Berlin. https://doi.org/10.1007/978-0-85729-021-2
[9]
Hasselblatt, B. and Katok, A. (2003) A First Course in Dynamics. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511998188
[10]
Gillman, L. and Jerison, M. (1960) Rings of Continuous Functions. V. Dan Nostrand Company, Inc., Princeton. https://doi.org/10.1007/978-1-4615-7819-2
[11]
Kechris, S.A. (2010) Global Aspects of Ergodic Group Actions. Americal Mathematical Society, Mathematical Surveys and Monographs, 160. https://doi.org/10.1090/surv/160
[12]
Mackenzie, K.C. (2005) General Theory of Lie Groupoids and Lie Algebroids. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9781107325883
[13]
Semadeni, Z. (1965) Spaces of Continuous Functions on Compact Sets. Advances in Mathematics, 1, 319-382. https://doi.org/10.1016/0001-8708(65)90041-1
[14]
Hahn, P. (1978) Haar Measure for Meaure Groupoids. Transactions of the American Mathematical Society, 242, 1-33. https://doi.org/10.1090/S0002-9947-1978-0496796-6
[15]
Muhly, P.S., Renault, J.N. and Williams, D.P. (1987) Equivalence and Isomorphism for Groupoid C*-Algebras. Journal of Operator Theory, 17, 3-22.
[16]
Okeke, O.N. and Egwe, M.E. (2018) Groupoid in the Analysis of Cohomogeneity One G-Spaces. Far East Journal of Mathematics, 103, 1903-1920. https://doi.org/10.17654/MS103121903
[17]
Deitmar, A. (2016) On Haar Systems for Groupoids. arXiv: 1605.08580v2.
[18]
Seda, K.A. (1986) On the Continuity of Haar Measure on Topological Groupoids. Proceedings of the American Mathematical Society, 96, 115-120. https://doi.org/10.2307/2045664
