Geometry of Stiefel manifolds
Author:
Eduardo Chiumiento
Journal:
Proc. Amer. Math. Soc. 138 (2010), 341353
MSC (2000):
Primary 22E65; Secondary 47B10, 58B20
Published electronically:
August 28, 2009
MathSciNet review:
2550200
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Abstract: Let be a separable Banach ideal in the space of bounded operators acting in a Hilbert space and the BanachLie group of unitary operators which are perturbations of the identity by elements in . In this paper we study the geometry of the unitary orbits and where is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in , and while the first one consists of partial isometries with the same kernel of , the second is given by partial isometries such that their initial projections and have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space and homogeneous reductive spaces of and respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of (or ) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.
 1.
Esteban
Andruchow and Gustavo
Corach, Metrics in the set of partial isometries with finite
rank, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei
(9) Mat. Appl. 16 (2005), no. 1, 31–44
(English, with English and Italian summaries). MR 2225921
(2007b:47185)
 2.
Esteban
Andruchow and Gustavo
Corach, Differential geometry of partial isometries and partial
unitaries, Illinois J. Math. 48 (2004), no. 1,
97–120. MR
2048217 (2005c:46074)
 3.
E.
Andruchow, G.
Corach, and M.
Mbekhta, On the geometry of generalized inverses, Math. Nachr.
278 (2005), no. 78, 756–770. MR 2141955
(2006f:46047), http://dx.doi.org/10.1002/mana.200310270
 4.
J.
Avron, R.
Seiler, and B.
Simon, The index of a pair of projections, J. Funct. Anal.
120 (1994), no. 1, 220–237. MR 1262254
(95b:47012), http://dx.doi.org/10.1006/jfan.1994.1031
 5.
Daniel
Beltiţă, Smooth homogeneous structures in operator
theory, Chapman & Hall/CRC Monographs and Surveys in Pure and
Applied Mathematics, vol. 137, Chapman & Hall/CRC, Boca Raton, FL,
2006. MR
2188389 (2007c:58010)
 6.
Daniel
Beltiţă, Tudor
S. Ratiu, and Alice
Barbara Tumpach, The restricted Grassmannian, Banach LiePoisson
spaces, and coadjoint orbits, J. Funct. Anal. 247
(2007), no. 1, 138–168. MR 2319757
(2009d:58014), http://dx.doi.org/10.1016/j.jfa.2007.03.001
 7.
A.
L. Carey, Some homogeneous spaces and representations of the
Hilbert Lie group 𝒰(ℋ)₂, Rev. Roumaine Math.
Pures Appl. 30 (1985), no. 7, 505–520. MR 826232
(87e:22044)
 8.
I.
C. Gohberg and M.
G. Kreĭn, Introduction to the theory of linear
nonselfadjoint operators, Translated from the Russian by A. Feinstein.
Translations of Mathematical Monographs, Vol. 18, American Mathematical
Society, Providence, R.I., 1969. MR 0246142
(39 #7447)
 9.
P.
R. Halmos and J.
E. McLaughlin, Partial isometries, Pacific J. Math.
13 (1963), 585–596. MR 0157241
(28 #477)
 10.
Serge
Lang, Introduction to differentiable manifolds, 2nd ed.,
Universitext, SpringerVerlag, New York, 2002. MR 1931083
(2003h:58002)
 11.
Luis
E. MataLorenzo and Lázaro
Recht, Infinitedimensional homogeneous reductive spaces, Acta
Cient. Venezolana 43 (1992), no. 2, 76–90
(English, with English and Spanish summaries). MR 1185114
(93j:46052)
 12.
Mostafa
Mbekhta and Şerban
Strǎtilǎ, Homotopy classes of partial isometries in
von Neumann algebras, Acta Sci. Math. (Szeged) 68
(2002), no. 12, 271–277. MR 1916580
(2003f:46097)
 13.
Iain
Raeburn, The relationship between a commutative Banach algebra and
its maximal ideal space, J. Functional Analysis 25
(1977), no. 4, 366–390. MR 0458180
(56 #16383)
 14.
Ioana
Serban and Flavius
Turcu, Compact perturbations of
isometries, Proc. Amer. Math. Soc.
135 (2007), no. 4,
1175–1180 (electronic). MR 2262923
(2007j:47023), http://dx.doi.org/10.1090/S0002993906085868
 15.
Şerban
Strătilă and Dan
Voiculescu, On a class of KMS states for the unitary group
𝑈(∞), Math. Ann. 235 (1978),
no. 1, 87–110. MR 0482248
(58 #2327)
 1.
 E. Andruchow, G. Corach, Metrics in the set of partial isometries with finite rank, Rendiconti Matematici della Accademia Nazionale dei Lincei s. 9, 16 (2005), 3144. MR 2225921 (2007b:47185)
 2.
 E. Andruchow, G. Corach, Differential geometry of partial isometries and partial unitaries, Illinois J. Math. 48 (2004), no. 1, 97120. MR 2048217 (2005c:46074)
 3.
 E. Andruchow, G. Corach, M. Mbekhta, On the geometry of generalized inverses, Math. Nachr. 278 (2005), no. 78, 756770. MR 2141955 (2006f:46047)
 4.
 J. Avron, R. Seiler, B. Simon, The index of a pair of projections, J. Funct. Anal. 120 (1994), no. 1, 220237. MR 1262254 (95b:47012)
 5.
 D. Beltiţă, Smooth homogeneous structures in operator theory, Monographs and Surveys in Pure and Applied Mathematics, 137, Chapman and Hall/CRC, Boca Raton, FL, 2006. MR 2188389 (2007c:58010)
 6.
 D. Beltiţă, T. S. Ratiu, A. B. Tumpach, The restricted Grassmannian, Banach LiePoisson spaces, and coadjoint orbits, J. Funct. Anal. 247 (2007), no. 1, 138168. MR 2319757 (2009d:58014)
 7.
 A. L. Carey, Some homogeneous spaces and representations of the Hilbert Lie group , Rev. Roumaine Math. Pures Appl. 30 (1985), no. 7, 505520. MR 826232 (87e:22044)
 8.
 I. C. Gohberg, M. G. Kreın, Introduction to the theory of linear nonselfadjoint operators, Amer. Math. Soc., Providence, RI, 1969. MR 0246142 (39:7447)
 9.
 P. R. Halmos, J. E. McLaughlin, Partial isometries, Pacific J. Math. 13 (1963), 585596. MR 0157241 (28:477)
 10.
 S. Lang, Introduction to differentiable manifolds. Second edition. Universitext, SpringerVerlag, New York, 2002. MR 1931083 (2003h:58002)
 11.
 L. MataLorenzo, L. Recht, Infinitedimensional homogeneous reductive spaces, Acta Cient. Venezolana 43 (1992), 7690. MR 1185114 (93j:46052)
 12.
 M. Mbekhta, Ş. Strătilă, Homotopy classes of partial isometries in von Neumann algebras, Acta Sci. Math. (Szeged) 68 (2002), 271277. MR 1916580 (2003f:46097)
 13.
 I. Raeburn, The relationship between a commutative Banach algebra and its maximal ideal space, J. Funct. Anal. 25 (1977), no. 4, 366390. MR 0458180 (56:16383)
 14.
 I. Serban, F. Turcu, Compact perturbations of isometries, Proc. Amer. Math. Soc. 135 (2007), 11751180. MR 2262923 (2007j:47023)
 15.
 Ş. Strătilă, D. Voiculescu, On a class of KMS states for the group , Math. Ann. 235 (1978), 87110. MR 0482248 (58:2327)
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Additional Information
Eduardo Chiumiento
Affiliation:
Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Calles 50 y 115, (1900) La Plata, Argentina
Email:
eduardo@mate.unlp.edu.ar
DOI:
http://dx.doi.org/10.1090/S0002993909100801
PII:
S 00029939(09)100801
Keywords:
Partial isometry,
Banach ideal,
Finsler metric
Received by editor(s):
September 18, 2008
Received by editor(s) in revised form:
April 22, 2009
Published electronically:
August 28, 2009
Communicated by:
Marius Junge
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
