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2005 | 3 | 2 | 188-202
Tytuł artykułu

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

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Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
2
Strony
188-202
Opis fizyczny
Daty
wydano
2005-06-01
online
2005-06-01
Twórcy
autor
Bibliografia
  • [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/s002090050496
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  • [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.
  • [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/BF01112353
  • [5] G. Horn: “Characterization of the predual and ideal structure of a JBW*-triple”, Math. Scan., Vol. 61, (1987), pp. 117–133.
  • [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.
  • [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.
  • [8] J.M. Isidro and L. L. Stachó: “On the manifold of finite rank tripotents in JB*-triples”, J. Math. Anal. Appl., to appear.
  • [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/BF01173928
  • [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/BF01465868
  • [11] W. Kaup: “On Grassmannians associated with JB*-triples”, Math. Z., Vol. 236, (2001), pp. 567–584. http://dx.doi.org/10.1007/PL00004842
  • [12] O. Loos: Bounded symmetric domains and Jordan pairs Mathematical Lectures, University of California at Irvine, 1977.
  • [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/04510037
  • [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/BF02108300
  • [15] H. Upmeier: Symmetric Banach manifolds and Jordan C *-algebras, North Holland Math. Studies, Vol. 104, Amsterdam, 1985.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_BF02479195
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