Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Finsler geometry and actions of the $ p$-Schatten unitary groups

Authors: Esteban Andruchow, Gabriel Larotonda and Lázaro Recht
Journal: Trans. Amer. Math. Soc. 362 (2010), 319-344
MSC (2000): Primary 22E65; Secondary 58B20, 58E50
Published electronically: July 27, 2009
MathSciNet review: 2550153
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p$ be an even positive integer and $ U_p(\mathcal{H})$ the Banach-Lie group of unitary operators $ u$ which verify that $ u-1$ belongs to the $ p$-Schatten ideal $ \mathcal{B}_p(\mathcal{H})$. Let $ \mathcal{O}$ be a smooth manifold on which $ U_p(\mathcal{H})$ acts transitively and smoothly. Then one can endow $ \mathcal{O}$ with a natural Finsler metric in terms of the $ p$-Schatten norm and the action of $ U_p(\mathcal{H})$. Our main result establishes that for any pair of given initial conditions

$\displaystyle x\in\mathcal{O}\hbox{ and } X\in(T \mathcal{O})_x $

there exists a curve $ \delta(t)=e^{tz}\cdot x$ in $ \mathcal{O}$, with $ z$ a skew-hermitian element in the $ p$-Schatten class, such that

$\displaystyle \delta(0)=x \hbox{ and } \dot{\delta}(0)=X, $

which remains minimal as long as $ t\Vert z\Vert _p\le \pi/4$. Moreover, $ \delta$ is unique with these properties. We also show that the metric space $ (\mathcal{O},d)$ (where $ d$ is the rectifiable distance) is complete. In the process we establish minimality results in the groups $ U_p(\mathcal{H})$ and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit

$\displaystyle \mathcal{O}=\{uAu^*: u\in U_p(\mathcal{H})\} $

of a self-adjoint operator $ A\in\mathcal{B}(\mathcal{H})$.

References [Enhancements On Off] (What's this?)

  • 1. E. Andruchow, G. Larotonda, Hopf-Rinow theorem in the Sato Grassmannian, J. Funct. Anal. 255 (2008), no. 7, 1692-1712. MR 2442079
  • 2. E. Andruchow, L. Recht, Geometry of unitaries in a finite algebra: variation formulas and convexity, Int. J. Math (to appear).
  • 3. E. Andruchow, D. Stojanoff, Geometry of unitary orbits, J. Operator Theory 26 (1991), no. 1, 25-41. MR 1214918 (94f:46068)
  • 4. C. Apostol, L. A. Fialkow, D. A. Herrero, D. V. Voiculescu, Approximation of Hilbert space operators. Vol. II, Research Notes in Mathematics, 102, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 735080 (85m:47002)
  • 5. D. Beltiţ$ \check{a}$, Smooth homogeneous structures in operator theory, Chapman et Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 137, Chapman et Hall/CRC, Boca Raton, FL, 2006. MR 2188389 (2007c:58010)
  • 6. D. Beltiţ$ \breve{a}$, T. S. Ratiu, A. B. Tumpach, The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits, J. Functional Analysis 247 (2007), no. 1, 138-168. MR 2319757
  • 7. P. Bona, Some considerations on topologies of infinite dimensional unitary coadjoint orbits, J. Geom. Phys. 51 (2004), no. 2, 256-268. MR 2078674 (2005f:58004)
  • 8. A. L. Carey, Some homogeneous spaces and representations of the Hilbert Lie group $ U(\mathcal{H})_2$, Rev. Roumaine Math. Pures Appl. 30 (1985), no. 7, 505-520. MR 826232 (87e:22044)
  • 9. C. Li, An Estimate for Lipschitz Constants of Metric Projections, J. Math. Anal. Appl. 231 (1999), no. 1, 133-141. MR 1676721 (99m:41053)
  • 10. G. Corach, H. Porta and L. Recht, The geometry of spaces of projections in $ C^*$-algebras, Adv. in Math. 41 (1997), no. 1, 54-76.
  • 11. C. E. Durán, L. E. Mata-Lorenzo, L. Recht, Natural variational problems in the Grassmann manifold of a C$ ^*$-algebra with trace, Adv. Math. 154 (2000), 196-228. MR 1780098 (2002e:58012)
  • 12. P. de la Harpe, Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space, Lecture Notes in Mathematics 285, Springer, Berlin, 1972. MR 0476820 (57:16372)
  • 13. L. A. Fialkow, A note on norm ideals and the operator $ X\longrightarrow AX-XB$, Israel J. Math. 32 (1979), no. 4, 331-348. MR 571087 (81g:47046)
  • 14. D. A. Herrero, Approximation of Hilbert space operators. Vol. 1. Second edition. Pitman Research Notes in Mathematics Series, 224, Longman Scientific & Technical, Harlow; co-published in the United States with John Wiley & Sons, Inc., New York, 1989. MR 1088255 (91k:47002)
  • 15. S. Lang, Differential and Riemannian manifolds, Third edition. Graduate Texts in Mathematics, 160, Springer-Verlag, New York, 1995. MR 1335233 (96d:53001)
  • 16. G. Larotonda, Unitary orbits in a full matrix algebra, Integ. Equat. Oper. Th. 54 (2006), no. 4, 511-523. MR 2222981 (2007c:58008)
  • 17. L. Mata-Lorenzo and L. Recht, Infinite dimensional homogeneous reductive spaces, Acta Cient. Venezolana 43 (1992), no. 2, 76-90. MR 1185114 (93j:46052)
  • 18. L.E. Mata-Lorenzo, L. Recht, Convexity properties of $ Tr[(a^*a)^n]$, Linear Alg. Appl. 315 (2000), 25-38. MR 1774958 (2002h:47116)
  • 19. H. Porta and L. Recht, Minimality of geodesics in Grassmann manifolds, Proc. Amer. Math. Soc. 100 (1987), no. 3, 464-466. MR 891146 (88f:46113)
  • 20. I. 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)
  • 21. G. Segal, G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. No. 61 (1985), 5-65. MR 783348 (87b:58039)
  • 22. R. W. Sharpe, Differential geometry. Cartan's generalization of Klein's Erlangen program. With a foreword by S. S. Chern, Graduate Texts in Mathematics 166, Springer-Verlag, New York, 1997. MR 1453120 (98m:53033)
  • 23. B. Simon, Trace ideals and their applications, Second edition. Mathematical Surveys and Monographs 120, American Mathematical Society, Providence, RI, 2005. MR 2154153 (2006f:47086)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 22E65, 58B20, 58E50

Retrieve articles in all journals with MSC (2000): 22E65, 58B20, 58E50

Additional Information

Esteban Andruchow
Affiliation: Instituto de Ciencias, J. M. Gutierrez 1150, (1613) Los Polvorines, Buenos Aires, Argentina

Gabriel Larotonda
Affiliation: Instituto de Ciencias, J. M. Gutierrez 1150, (1613) Los Polvorines, Buenos Aires, Argentina

Lázaro Recht
Affiliation: Departamento de Matemática P y A, Universidad Simón Bolívar, Apartado 89000, Caracas 1080A, Venezuela

Received by editor(s): December 19, 2007
Published electronically: July 27, 2009
Additional Notes: This work was partially supported by IAM-CONICET
Dedicated: In memory of A. R. Larotonda (1939-2005)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society