Direct integral of locally Hilbert spaces

2025/08/05 by Kulkarni, Chaitanya J., Pamula, Santhosh Kumar
#46A03 #46A13 #46A40 #46L05 #47L40 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2508.03407

Abstract

In this work, we introduce the concept of the direct integral of locally Hilbert spaces. This notion is formulated such that the direct integral of locally Hilbert spaces forms a locally Hilbert space. We then define two classes of locally bounded operators on the direct integral of locally Hilbert spaces namely the class of decomposable locally bounded operators and the class of diagonalizable locally bounded operators. We prove that the set of all decomposable locally bounded operators forms a locally von Neumann algebra, while the set of diagonalizable locally bounded operators forms an abelian von Neumann algebra, and we show that they are commutant of each other.

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 over a locally measure space
• Direct Integral and Decompoisitions of Locally Hilbert spaces
• Direct integrals of strongly irreducible operators
• Fuglede-Putnam theorem for locally measurable operators
• Operator Models for Hilbert Locally C^*-Modules