Abstract and Applied Analysis

On the Geometry of the Unit Ball of a J B * -Triple

Haifa M. Tahlawi, Akhlaq A. Siddiqui, and Fatmah B. Jamjoom

Full-text: Open access


We explore a J B * -triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting of C * -algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in a J B * -triple; this indicates their structural richness. We initiate a study of the unit ball of a J B * -triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. Some C * -algebra and J B * -algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended to J B * -triples.

Article information

Abstr. Appl. Anal., Volume 2013 (2013), Article ID 891249, 8 pages.

First available in Project Euclid: 27 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Tahlawi, Haifa M.; Siddiqui, Akhlaq A.; Jamjoom, Fatmah B. On the Geometry of the Unit Ball of a $J{B}^{\mathrm{*}}$ -Triple. Abstr. Appl. Anal. 2013 (2013), Article ID 891249, 8 pages. doi:10.1155/2013/891249. https://projecteuclid.org/euclid.aaa/1393511914

Export citation


  • L. G. Brown and G. K. Pedersen, “On the geometry of the unit ball of a ${C}^{\ast\,\!}$-algebra,” Journal für die Reine und Angewandte Mathematik, vol. 469, pp. 113–147, 1995.
  • L. G. Brown and G. K. Pedersen, “Approximation and convex decomposition by extremals in a ${C}^{\ast\,\!}$-algebra,” Mathematica Scandinavica, vol. 81, no. 1, pp. 69–85, 1997.
  • H. Upmeier, Symmetric Banach manifolds and Jordan C$^{\ast\,\!}$-algebras, vol. 104 of North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam, The Netherlands, 1985.
  • H. M. Tahlawi and A. A. Siddiqui, “Analogue of Brown-Pedersen' quasi invertibility for $J{B}^{\ast\,\!}$-triples,” International Journal of Mathematical Analysis, vol. 5, no. 30, pp. 1469–1476, 2011.
  • H. M. Tahlawi and A. A. Siddiqui, “On non-degenerate Jordan triple systems,” International Journal of Algebra, vol. 5, no. 21-24, pp. 1099–1105, 2011.
  • J. D. M. Wright and M. A. Youngson, “A Russo-Dye theorem for Jordan ${C}^{\ast\,\!}$-algebras,” in Functional Analysis: Surveys and Recent Results, vol. 27 of Mathematical Studies, pp. 279–282, North-Holland, Amsterdam, The Netherlands, 1977.
  • A. A. Siddiqui, “A proof of the Russo-Dye theorem for $J{B}^{\ast\,\!}$-algebras,” New York Journal of Mathematics, vol. 16, pp. 53–60, 2010.
  • W. Kaup, “A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces,” Mathematische Zeit-schrift, vol. 183, no. 4, pp. 503–529, 1983.
  • M. Burgos, F. J. Fernández-Polo, J. J. Garcés, and A. M. Peralta, “Orthogonality preservers revisited,” Asian-European Journal of Mathematics, vol. 2, no. 3, pp. 387–405, 2009.
  • W. Kaup, “On Grassmannians associated with $J{B}^{\ast\,\!}$-triples,” Mathematische Zeitschrift, vol. 236, no. 3, pp. 567–584, 2001.
  • W. Kaup, “On spectral and singular values in $J{B}^{\ast\,\!}$-triples,” Proceedings of the Royal Irish Academy. Section A, vol. 96, no. 1, pp. 95–103, 1996.
  • O. Loos, Jordan Pairs, Lecture Notes in Mathematics, Springer, New York, NY, USA, 1975.
  • K. Meyberg, “von Neumann regularity in Jordan triple systems,” Archiv der Mathematik, vol. 23, pp. 589–593, 1972.
  • W. Kaup, “On the CR-structure of certain linear group orbits in infinite dimensions,” Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, vol. 3, no. 3, pp. 535–554, 2004.
  • A. Fernández López, E. García Rus, E. Sánchez Campos, and M. Siles Molina, “Strong regularity and generalized inverses in Jordan systems,” Communications in Algebra, vol. 20, no. 7, pp. 1917–1936, 1992.
  • L. J. Bunce, C. H. Chu, and B. Zalar, “Structure spaces and decomposition in $J{B}^{\ast\,\!}$-triples,” Mathematica Scandinavica, vol. 86, no. 1, pp. 17–35, 2000.
  • C. M. Edwards and G. T. Rüttimann, “Compact tripotents in bi-dual $J{B}^{\ast\,\!}$-triples,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 120, no. 1, pp. 155–173, 1996.
  • L. L. Stachó, “On the spectrum of inner derivations in partial Jordan triples,” Mathematica Scandinavica, vol. 66, no. 2, pp. 242–248, 1990.
  • M. Burgos, E. A. Kaidi, A. M. Campoy, A. M. Peralta, and M. Ramírez, “Von Neumann regularity and quadratic conorms in $J{B}^{\ast\,\!}$-triples and ${C}^{\ast\,\!}$-algebras,” Acta Mathematica Sinica, vol. 24, no. 2, pp. 185–200, 2008.
  • A. A. Siddiqui, “Average of two extreme points in $JB{W}^{\ast\,\!}$-triples,” Japan Academy. Proceedings. Series A, vol. 83, no. 9-10, pp. 176–178, 2007.
  • M. A. Rakha, “On the Moore-Penrose generalized inverse matrix,” Applied Mathematics and Computation, vol. 158, no. 1, pp. 185–200, 2004.
  • A. A. Siddiqui, “$J{B}^{\ast\,\!}$-algebras of topological stable rank 1,” International Journal of Mathematics and Mathematical Sciences, Article ID 37186, 24 pages, 2007.
  • W. Kaup and H. Upmeier, “Jordan algebras and symmetric Siegel domains in Banach spaces,” Mathematische Zeitschrift, vol. 157, no. 2, pp. 179–200, 1977.
  • A. A. Siddiqui, “Self-adjointness in unitary isotopes of $J{B}^{\ast\,\!}$-alge-bras,” Archiv der Mathematik, vol. 87, no. 4, pp. 350–358, 2006.
  • J. M. Isidro, “A glimpse at the theory of Jordan-Banach triple systems,” Revista Matemática de la Universidad Complutense de Madrid, vol. 2, pp. 145–156, 1989.
  • A. A. Siddiqui, “Convex combinations of unitaries in $J{B}^{\ast\,\!}$-alge-bras,” New York Journal of Mathematics, vol. 17, pp. 127–137, 2011.
  • M. Rørdam, “Advances in the theory of unitary rank and regular approximation,” Annals of Mathematics, vol. 128, no. 1, pp. 153–172, 1988.
  • A. A. Siddiqui, “The ${\lambda }_{u}$-function in $J{B}^{\ast\,\!}$-algebras,” New York Journal of Mathematics, vol. 17, pp. 139–147, 2011.