Abelian locally von Neumann algebras acting on separable locally Hilbert spaces with sequentially finite index sets are spatially isomorphic to the algebra of all locally diagonalizable operators on a direct integral of locally Hilbert spaces.
Dosi , A survey of multinormed von Neumann algebras, Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 43(2017), 3--67
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Functional Models of Abelian Locally von Neumann Algebras and Direct Integrals of Locally Hilbert Spaces
Abelian locally von Neumann algebras acting on separable locally Hilbert spaces with sequentially finite index sets are spatially isomorphic to the algebra of all locally diagonalizable operators on a direct integral of locally Hilbert spaces.