Direct Integral and Decompoisitions of Locally Hilbert spaces

Abstract

In this work, we introduce the concept of direct integral of locally Hilbert spaces by using the notion of locally standard measure space (analogous to standard measure space defined in the classical setup), which we obtain by considering a strictly inductive system of measurable spaces along with a projective system of finite measures. Next, we define a locally Hilbert space given by the direct integral of a family of locally Hilbert spaces. Following that we introduce decomposable locally bounded and diagonalizable locally bounded operators. Further, we show that the class of diagonalizable locally bounded operators is an abelian locally von Neumann algebra, and this can be seen as the commutant of decomposable locally bounded operators. Finally, we discuss the following converse question: For a locally Hilbert space $\mathcal{D}$ and an abelian locally von Neumann algebra $\mathcal{M}$, does there exist a locally standard measure space and a family of locally Hilbert spaces such that (1) the locally Hilbert space $\mathcal{D}$ is identified with the direct integral of family of locally Hilbert spaces; (2) the abelian locally von Neumann algebra $\mathcal{M}$ is identified with the abelian locally von Neumann algebra of all diagonalizable locally bounded operators ? We answer this question affirmatively for a certain class of abelian locally von Neumann algebras.