Abstract
Let H be a Hilbert space and A ⊆ B(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 p ∈ GP can be joined with p by a unique geodesic curve in GP.
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References
Akhiezer N I, Glazman I M. Theory of Linear Operators in Hilbert Space. New York: Dover Publications, 1993
Andruchow E. The Grassmann manifold of a Hilbert space. Actas del Xll Congreso Dr Antonio A R Monteiro. Http://inmabb-conicet.gob.ar/static/publicaciones/actas/12/08-andruchow.pdf, 2014, 41–55
Andruchow E. Pairs of projections: Geodesics, Fredholm and compact pairs. Complex Anal Oper Theory, 2014, 8: 1435–1453
Andruchow E, Corach G. Differential geometry of partial isometries and partial unitaries. Illinois J Math, 2004, 48: 97–120
Andruchow E, Corach G, Mbekhta M. On the geometry of generalized inverses. Math Nachr, 2005, 278: 756–770
Andruchow E, Corach G, Mbekhta M. A note on the differentiable structure of generalized idempotents. Cent Eur J Math, 2013, 11: 1004–1019
Andruchow E, Corach G, Stojanoff D. Projective spaces of a C*-algebra. Integral Equations Operator Theory, 2000, 37: 143–168
Apostol C, Fialkow L A, Herrero D A, et al. Approximation of Hilbert Space Operators, Volume II. London: Pitman Advanced Pub, 1984
Avron J, Seiler R, Simon B. The index of a pair of projections. J Funct Anal, 1994, 120: 220–237
Corach G, Porta H, Recht L. The geometry of spaces of projections in C*-algebras. Adv Math, 1993, 101: 27–34
Deng C Y, Du H K. Common complements of two subspaces and an answer to Groß’s question. Acta Math Sin Engl Ser, 2006, 49: 1099–1112
Du H K, Li Y. The spectral characterization of generalized projections. Linear Algebra Appl, 2005, 400: 313–318
Halmos P R, Mclaughin J E. Partial isometries. Pacic J Math, 1963, 12: 585–596
Kato T. Notes on Projections and Perturbation Theory. Technical Report No. 9. California: University of California, 1955
Kato T. Perturbation Theory for Linear Operators. Berlin-Heidelberg: Springer-Verlag, 1966
Lang S. Differential Manifolds. Upper Saddle River: Addison-Wesley, 1972
Massera J L, Schäffer J J. Linear Differential Equations and Function Spaces. New York: Academic Press, 1966
Porta H, Recht L. Minimality of geodesic in Grassmann manifolds. Proc Amer Math Soc, 1987, 100: 464–466
Porta H, Recht L. Spaces of projections in a Banach algebra. Acta Cient Venezolana, 1987, 38: 408–426
Upmeier H. Symmetric Banach Manifolds and Jordan C*-Algebras. Amsterdam: North-Holland, 1985
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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Cui, M., Ji, G. On geometric structure of generalized projections in C*-algebras. Sci. China Math. 61, 1187–1200 (2018). https://doi.org/10.1007/s11425-017-9111-7
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DOI: https://doi.org/10.1007/s11425-017-9111-7