$L_1$-space for a positive operator affiliated with von Neumann algebra

TL;DR

This paper develops an L1-type space for positive selfadjoint operators affiliated with von Neumann algebras, establishing its properties, representations, and connections to preduals and weights, extending to C*-algebras.

Contribution

It introduces a new L1-type space for such operators, proves its isometric isomorphism to the predual, and explores its properties and connections to weights and C*-algebras.

Findings

L1-space is a norm if and only if the operator is injective

L1-space is isometrically isomorphic to the predual of the von Neumann algebra

Connections established between semifinite weights and positive elements in the L1-space

Abstract

In this paper we suggest an approach for constructing an L1-type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we intro- duce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an L1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent L1- type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that L1 -type space is isometrically isomorphic to the predual of von Neumann algebra in a natural way. We give a small list of alternate definitions of the seminorm, and a special definition for the case of semifinite von Neumann algebra, in particular. We study order properties of L1-type space, and demonstrate the con- nection between semifinite normal weights and positive elements of this space. At last, we construct a similar L-space for the positive element of C*-algebra, and study the connection between this L-space and the L1 -type space in case when this C*-algebra is a von Neumann algebra.