On geometric structure of generalized projections in C*-algebras

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Abstract

Let H be a Hilbert space and AB(H) a C*-subalgebra. This paper is devoted to studying the set GP of generalized projections in A from a differential geometric point of view, and mainly focuses on geodesic curves. We prove that the space GP is a C Banach submanifold of A, and a homogeneous reductive space under the action of Banach Lie group U A of A. Moreover, we compute the geodesics of GP in a standard fashion, and prove that any generalized projection in a prescribed neighbourhood of pGP can be joined with p by a unique geodesic curve in GP.

Keywords

generalized projections Banach manifold geodesics 

MSC(2010)

32M10 53C22 53C30 47C15 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11371233). The authors thank the referees for their valuable comments and suggestions.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina

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