Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2016: 0.454
Source Normalized Impact per Paper (SNIP) 2016: 0.850

Mathematical Citation Quotient (MCQ) 2016: 0.23

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 3, Issue 2 (Jun 2005)

Issues

Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ (H) of bounded linear operatorsof bounded linear operators

José Isidro
Published Online: 2005-06-01 | DOI: https://doi.org/10.2478/BF02479195

Abstract

Given a complex Hilbert space H, we study the manifold $$\mathcal{A}$$ of algebraic elements in $$Z = \mathcal{L}\left( H \right)$$ . We represent $$\mathcal{A}$$ as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M, and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection.

Keywords: Jordan-Banach algebras; JB*-triples; algebraic elements; Grassmann manifolds; Riemann manifolds

Keywords: 17C27; 17C36; 17B65

  • [1] C.H. Chu and J.M. Isidro: “Manifolds of tripotents in JB*-triples”, Math. Z., Vol. 233, (2000), pp. 741–754. http://dx.doi.org/10.1007/s002090050496CrossrefGoogle Scholar

  • [2] S. Dineen: The Schwarz lemma, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1989. Google Scholar

  • [3] L.A. Harris: “Bounded symmetric homogeneous domains in infinite dimensional spaces”, In: Proceedings on Infinite dimensional Holomorphy, Lecture Notes in Mathematics, Vol. 364, 1973, Springer-Verlag, Berlin, 1973, pp. 13–40. Google Scholar

  • [4] U. Hirzebruch: “Über Jordan-Algebren und kompakte Riemannsche symmetrische Räume von Rang 1”, Math. Z., Vol. 90, (1965), pp. 339–354. http://dx.doi.org/10.1007/BF01112353CrossrefGoogle Scholar

  • [5] G. Horn: “Characterization of the predual and ideal structure of a JBW*-triple”, Math. Scan., Vol. 61, (1987), pp. 117–133. Google Scholar

  • [6] J.M. Isidro: The manifold of minimal partial isometries in the space \(\mathcal{L}\) (H,K) of bounded linear operators”, Acta Sci. Math. (Szeged), Vol. 66, (2000), pp. 793–808. Google Scholar

  • [7] J.M. Isidro and M. Mackey: “The manifold of finite rank projections in the algebra \(\mathcal{L}\) (H) of bounded linear operators”, Expo. Math., Vol. 20 (2), (2002), pp. 97–116. CrossrefGoogle Scholar

  • [8] J.M. Isidro and L. L. Stachó: “On the manifold of finite rank tripotents in JB*-triples”, J. Math. Anal. Appl., to appear. Google Scholar

  • [9] W. Kaup: “A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces”, Math. Z., Vol. 183, (1983), pp. 503–529. http://dx.doi.org/10.1007/BF01173928CrossrefGoogle Scholar

  • [10] W. Kaup: “Über die Klassifikation der symmetrischen Hermiteschen Mannigfaltigkeiten unendlicher Dimension, I, II”, Math. Ann., Vol. 257, (1981), pp. 463–483 and Vol. 262, (1983), pp. 503–529. http://dx.doi.org/10.1007/BF01465868CrossrefGoogle Scholar

  • [11] W. Kaup: “On Grassmannians associated with JB*-triples”, Math. Z., Vol. 236, (2001), pp. 567–584. http://dx.doi.org/10.1007/PL00004842CrossrefGoogle Scholar

  • [12] O. Loos: Bounded symmetric domains and Jordan pairs Mathematical Lectures, University of California at Irvine, 1977. Google Scholar

  • [13] T. Nomura: “Manifold of primitive idempotents in a Jordan-Hilbert algebra”, J. Math. Soc. Japan, Vol. 45, (1993), pp. 37–58. http://dx.doi.org/10.2969/jmsj/04510037CrossrefGoogle Scholar

  • [14] T. Nomura: “Grassmann manifold of a JH-algebra”, Annals of Global Analysis and Geometry, Vol. 12, (1994), pp. 237–260. http://dx.doi.org/10.1007/BF02108300CrossrefGoogle Scholar

  • [15] H. Upmeier: Symmetric Banach manifolds and Jordan C *-algebras, North Holland Math. Studies, Vol. 104, Amsterdam, 1985. Google Scholar

About the article

Published Online: 2005-06-01

Published in Print: 2005-06-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/BF02479195.

Export Citation

© 2005 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in