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.
Gheondea , On locally Hilbert spaces, Opuscula Mathematica 36(2016), 735--747
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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.