Direct integral of locally Hilbert spaces over a locally measure space

2025/08/05 by Kulkarni, Chaitanya J., Pamula, Santhosh Kumar
#46A03 #46A13 #46C05 #46M40 #47L10 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2508.03779

Abstract

In this work, we introduce the concept of the direct integral of locally Hilbert spaces by generalizing the classical notion of a measure space to that of a locally measure space. We establish that the direct integral of a family of locally Hilbert spaces over a locally measure space forms a locally Hilbert space. We then define two important subclasses of locally bounded operators on such direct integrals, namely decomposable locally bounded operators and diagonalizable locally bounded operators. We show that each of these subclasses forms a locally von Neumann algebra, and in particular, that the locally von Neumann algebra of diagonalizable operators is abelian. Finally, we prove that the locally von Neumann algebra of diagonalizable operators coincides with the commutant of the locally von Neumann algebra of decomposable operators.

Citations

• Stinespring's Theorem for Unbounded Operator valued Local completely positive maps and Its Applications
• Operator Models for Hilbert Locally C^*-Modules
• Local operator spaces, unbounded operators and multinormed C * -algebras

Related

• Direct integral of locally Hilbert spaces
• Direct Integral and Decompoisitions of Locally Hilbert spaces
• Fuglede-Putnam theorem for locally measurable operators
• Direct integrals of strongly irreducible operators
• Integral representation of translation-invariant operators on reproducing kernel Hilbert spaces