Advances in Pure Mathematics, 2019, 9, 879-924 https://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368 DOI: 10.4236/apm.2019.911044 Nov. 4, 2019 879 Advances in Pure Mathematics Disintegration of Group Representations on Direct Integrals of Banach Spaces Simon Joseph 1* , Fatin Saeed 2 , Nagat Suoliman 3 , Mohammed Khanfoor 4 , Abd Elmotaleb Abd Ellah 5 1 Department of Mathematics, College of Education, University of Bahr El-Ghazal, Wau, South Sudan 2 Department of Mathematics, University College at Qaryat Alulya, University of Hafr Albatin, Hafr Albatin, KSA 3 Department of Computer Science, College of Computer Science, King Khalid University, Alsamer Santer, KSA 4 Department of Mathematics, College of Education, Nile Valley University, Atbara, Sudan 5 Department of Mathematics, Faculty of Science, University of Kassala, Kassala, Sudan Abstract In this paper, let G be a Polish locally compact group acting on a Polish space X with a G-invariant probability measures j j µ ∑ . Factorize the integral with respect to j j µ ∑ in terms of the integrals with respect to the ergodic measures on X, and showed that ( ) ( ) ( ) 1 , ,0 j j L X µ + ≤ <∞ ∑   are G-equivar- iantly isometric ally lattice isomorphic to an ( ) 1 L + -direct integral of the spaces ( ) ( ) 1 , j L X λ + , where j λ ranges over the ergodic measures on X. This yields a disintegration of the canonical representation of G as isometric lattice auto orphisms of ( ) ( ) 1 , j j L X µ + ∑  as an ( ) 1 L + -direct integral of order in- decomposable representations. If ( ) , j j X µ ′ ′ ∑ are probability space, and, for some 0 ≤ <∞  , G acts in a strongly continuous manner on ( ) ( ) 1 , j j L X µ + ′ ′ ∑  as isometric lattice auto orphisms that leave the constants fixed, then G acts on ( ) ( ) 1 , j j L X µ + ′ ′ ∑  in a similar fashion for all 0 ≤ <∞  . Moreover, there exists an alternative model in which these representations originate from a continuous action of G on a compact Hausdorff space. If ( ) , j j X µ ′ ′ ∑ are separable, the representation of G on ( ) ( ) 1 , j j L X µ + ′ ′ ∑  can then be disintegrated into order indecomposable representations. The notions of ( ) 1 L + -direct integrals of Banach spaces and representation is de- veloped for extend those in the literature. How to cite this paper: Joseph, S., Saeed, F., Suoliman, N., Khanfoor, M. and Ellah, A.E.A. (2019) Disintegration of Group Repre- sentations on Direct Integrals of Banach Spaces. Advances in Pure Mathematics, 9, 879-924. https://doi.org/10.4236/apm.2019.911044 Received: July 11, 2019 Accepted: November 1, 2019 Published: November 4, 2019 Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution-NonCommercial International License (CC BY-NC 4.0). http://creativecommons.org/licenses/by-nc/4.0/ Open Access