$$L_1$$ L 1 -space for a positive operator affiliated with von Neumann algebra

$$L_1$$ L 1 -space for a positive operator affiliated with von Neumann algebra In this paper we suggest an approach for constructing an $$L_1$$ L 1 -type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we introduce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an $$L_1$$ L 1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent $$L_1$$ L 1 -type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that $$L_1$$ L 1 -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 $$L_1$$ L 1 -type space, and demonstrate the connection 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 $$L_1$$ L 1 -type space in case when this C*-algebra is a von Neumann algebra. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

$$L_1$$ L 1 -space for a positive operator affiliated with von Neumann algebra

Positivity , Volume 21 (1) – May 18, 2016
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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0422-4
Publisher site
See Article on Publisher Site

Abstract

In this paper we suggest an approach for constructing an $$L_1$$ L 1 -type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we introduce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an $$L_1$$ L 1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent $$L_1$$ L 1 -type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that $$L_1$$ L 1 -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 $$L_1$$ L 1 -type space, and demonstrate the connection 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 $$L_1$$ L 1 -type space in case when this C*-algebra is a von Neumann algebra.

Journal

PositivitySpringer Journals

Published: May 18, 2016

References

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