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Complete Descriptions of Intermediate Operator Algebras by Intermediate Extensions of Dynamical Systems

  • Yuhei SuzukiEmail author
Article
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Abstract

Practically and intrinsically, inclusions of operator algebras are of fundamental interest. The subject of this paper is intermediate operator algebras of inclusions. There are two previously known theorems which naturally and completely describe all intermediate operator algebras: the Galois Correspondence Theorem and the Tensor Splitting Theorem. Here we establish the third, new complete description theorem which gives a canonical bijective correspondence between intermediate operator algebras and intermediate extensions of dynamical systems. One can also regard this theorem as a crossed product splitting theorem, analogous to the Tensor Splitting Theorem. We then give concrete applications, particularly to maximal amenability problem and a new realization result of intermediate operator algebra lattice.

Notes

Acknowledgements

The author is grateful to Cyril Houdayer for a stimulating conversation on Margulis’ factor theorem [40] during his visiting at Université Paris-Sud in 2015. He is also grateful to Yoshimichi Ueda for helpful comments and discussions on the formulation of Theorem 4.6. He is also grateful to Toshihiko Masuda for letting him know the reference [62]. This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (Start-up, No. 17H06737) and tenure track funds of Nagoya University.

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

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan

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