arXiv subjects

Pierre de Jager

Publications and source records attributed to Pierre de Jager.

5 recordsLinked to original sources

arXiv2021

Isometries on non-commutative (quantum) Lorentz spaces associated with semi-finite von Neumann algebras

In this article we characterize the extreme points of the unit ball of a non-commutative (quantum) Lorentz space associated with a semi-finite von Neumann algebra. This enables us to show that surjective isometries between non-commutative Lorentz spaces are projection disjointness preserving and finiteness preserving, which facilitates a characterization of the structure of these isometries.

math.OA↗

arXiv2020

Extreme point methods in the study of isometries on certain non-commutative spaces

In this paper we characterize surjective isometries on certain classes of non-commutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$, as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$. The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.

math.OA↗

arXiv2019

Isometries between non-commutative symmetric spaces associated with semi-finite von Neumann algebras

In this article we show that positive surjective isometries between symmetric spaces associated with semi-finite von Neumann algebras are projection disjointness preserving if they are finiteness preserving. This is subsequently used to obtain a structural description of such isometries. Furthermore, it is shown that if the initial symmetric space is a strongly symmetric space with absolutely continuous norm, then a similar structural description can be obtained without requiring positivity of the isometry.

math.OA↗

arXiv2018

Extension of projection mappings

We show that a map between projection lattices of semi-finite von Neumann algebras can be extended to a Jordan $*$-homomorphism between the von Neumann algebras if this map is defined in terms of the support projections of images (under the linear map) of projections and the images of orthogonal projections have orthogonal support projections. This has numerous fundamental applications in the study of isometries and composition operators on quantum symmetric spaces and is of independent interest, since it provides a partial generalization of Dye's Theorem without the requirement that the initial von Neumann algebra be free of type $I_2$ summands.

math.OA↗

arXiv2018

Multiplication operators on non-commutative spaces

Boundedness and compactness properties of multiplication operators on quantum (non-commutative) function spaces are investigated. For endomorphic multiplication operators these properties can be characterized in the setting of quantum symmetric spaces. For non-endomorphic multiplication operators these properties can be completely characterized in the setting of quantum $L^p$-spaces and a partial solution obtained in the more general setting of quantum Orlicz spaces.

math.OA↗