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

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Abstract

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.

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Acknowledgements

The greater part of this research was conducted during the first author’s doctoral studies at the University of Cape Town. The first author would like to thank his Ph.D. supervisor, Dr. Robert Martin, for his input and guidance and the NRF for funding towards this project in the form of scarce skills and grantholder-linked bursaries. Furthermore, the support of the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the authors and are not necessarily attributed to the CoE. The authors would also like to thank the reviewer for pointing out the preprints [14, 15, 23] and for numerous useful comments and suggestions.

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de Jager, P., Conradie, J. Isometries between non-commutative symmetric spaces associated with semi-finite von Neumann algebras. Positivity 24, 815–835 (2020). https://doi.org/10.1007/s11117-019-00711-2

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