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Interpolation between \(L_0({\mathcal M},\tau )\) and \(L_\infty ({\mathcal M},\tau )\)

  • J. HuangEmail author
  • F. Sukochev
Article

Abstract

Let \({\mathcal M}\) be a semifinite von Neumann algebra with a faithful semifinite normal trace \(\tau \). We show that the symmetrically \(\Delta \)-normed operator space \(E({\mathcal M},\tau )\) corresponding to an arbitrary symmetrically \(\Delta \)-normed function space \(E(0,\infty )\) is an interpolation space between \(L_0({\mathcal M},\tau )\) and \({\mathcal M}\), which is in contrast with the classical result that there exist symmetric operator spaces \(E({\mathcal M},\tau )\) which are not interpolation spaces between \(L_1({\mathcal M},\tau )\) and \({\mathcal M}\). Besides, we show that the \({\mathcal K}\)-functional of every \(X\in L_0({\mathcal M},\tau )+{\mathcal M}\) coincides with the \({\mathcal K}\)-functional of its generalized singular value function \(\mu (X)\). Several applications are given, e.g., it is shown that the pair \((L_0({\mathcal M},\tau ),{\mathcal M})\) is \({\mathcal K}\)-monotone when \({\mathcal M}\) is a non-atomic finite factor.

Keywords

Interpolation Orbits \({\mathcal K}\)-orbits Symmetrically \(\Delta \)-normed spaces Von Neumann algebras 

Mathematics Subject Classification

46L10 46E30 47A57 

Notes

Acknowledgements

The authors would like to thank Sergei Astashkin and Dima Zanin for helpful discussions.

J. Huang acknowledges the support of University International Postgraduate Award (UIPA). F. Sukochev was supported by the Australian Research Council.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesKensingtonAustralia

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