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$JB^{\ast}$-algebras of topological stable rank 1. (English) Zbl 1161.46041
$C^*$-algebras of topological stable rank 1 were introduced by {\it M. A. Rieffel} [Proc. Lond. Math. Soc. (3) 46, 301--333 (1983; Zbl 0533.46046)]. Rieffel also proved that a $C^*$-algebra $A$ is of topological stable rank 1 (tsr 1) if and only if its invertible elements are norm dense. The paper under review considers this characterisation as starting point to extend the notion of topological stable rank 1 to the more general setting of $JB^*$-algebras. The author proves, among other results, that a complex spin factor and a finite-dimensional $JB^*$-algebra have stable rank one, providing an example of a special $JBW^*$-algebra of tsr 1 whose enveloping von Neumann algebra is not trs 1. The author also proves some interesting characterisations of invertible extreme points of the closed unit ball of a $JB^*$-algebra. Concretely, an extreme point of the closed unit ball of a unital $JB^*$-algebra is unitary if and only if its distance to the invertible elements is strictly less than 1. Consequently, given a $JB^*$-algebra of tsr 1 $A$, the extreme points of the closed unit ball of $A$ and unitary elements in $A$ coincide. This characterisation extends an analogous results obtained by {\it G. K. Pedersen} in [J. Oper. Theory 26, No. 2, 345--381 (1991; Zbl 0784.46043)].

46L70Nonassociative selfadjoint operator algebras
Full Text: DOI EuDML
[1] A. R. Pears, “Dimension Theory of General Spaces,” Cambridge University Press, Cambridge, UK, 1975. · Zbl 0312.54001
[2] M. A. Rieffel, “Dimension and stable rank in the K-theory of C\ast -algebras,” Proceedings of the London Mathematical Society. Third Series, vol. 46, no. 2, pp. 301-333, 1983. · Zbl 0533.46046 · doi:10.1112/plms/s3-46.2.301
[3] R. H. Herman and L. N. Vaserstein, “The stable range of C\ast -algebras,” Inventiones Mathematicae, vol. 77, no. 3, pp. 553-555, 1984. · Zbl 0559.46025 · doi:10.1007/BF01388839 · eudml:143161
[4] L. G. Brown and G. K. Pedersen, “C\ast -algebras of real rank zero,” Journal of Functional Analysis, vol. 99, no. 1, pp. 131-149, 1991. · Zbl 0776.46026 · doi:10.1016/0022-1236(91)90056-B
[5] R. V. Kadison and G. K. Pedersen, “Means and convex combinations of unitary operators,” Mathematica Scandinavica, vol. 57, no. 2, pp. 249-266, 1985. · Zbl 0573.46034 · eudml:166953
[6] G. K. Pedersen, “The \lambda -function in operator algebras,” Journal of Operator Theory, vol. 26, no. 2, pp. 345-381, 1991. · Zbl 0784.46043
[7] A. G. Robertson, “A note on the unit ball in C\ast -algebras,” Bulletin of the London Mathematical Society, vol. 6, no. 3, pp. 333-335, 1974. · Zbl 0291.46042 · doi:10.1112/blms/6.3.333
[8] M. Rørdam, “Advances in the theory of unitary rank and regular approximation,” Annals of Mathematics, vol. 128, no. 1, pp. 153-172, 1988. · Zbl 0659.46052 · doi:10.2307/1971465
[9] J. D. M. Wright, “Jordan C\ast -algebras,” The Michigan Mathematical Journal, vol. 24, no. 3, pp. 291-302, 1977. · Zbl 0384.46040 · doi:10.1307/mmj/1029001946
[10] A. A. Siddiqui, “Self-adjointness in unitary isotopes of JB\ast -algebras,” Archiv der Mathematik, vol. 87, no. 4, pp. 350-358, 2006. · Zbl 1142.46020 · doi:10.1007/s00013-006-1718-6
[11] R. M. Aron and R. H. Lohman, “A geometric function determined by extreme points of the unit ball of a normed space,” Pacific Journal of Mathematics, vol. 127, no. 2, pp. 209-231, 1987. · Zbl 0662.46020 · doi:10.2140/pjm.1987.127.209
[12] N. Jacobson, Structure and Representations of Jordan Algebras, vol. 39 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1968. · Zbl 0218.17010 · http://www.ams.org/online_bks/coll39/
[13] W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, NY, USA, 1973. · Zbl 0253.46001
[14] C. V. Devapakkiam, “Jordan algebras with continuous inverse,” Mathematica Japonica, vol. 16, pp. 115-125, 1971. · Zbl 0246.17015
[15] M. A. Youngson, “A Vidav theorem for Banach Jordan algebras,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 84, no. 2, pp. 263-272, 1978. · Zbl 0392.46038 · doi:10.1017/S0305004100055092
[16] E. M. Alfsen, F. W. Shultz, and E. Størmer, “A Gel/fand-Neumark theorem for Jordan algebras,” Advances in Mathematics, vol. 28, no. 1, pp. 11-56, 1978. · Zbl 0397.46065 · doi:10.1016/0001-8708(78)90044-0
[17] C. M. Glennie, “Some identities valid in special Jordan algebras but not valid in all Jordan algebras,” Pacific Journal of Mathematics, vol. 16, no. 1, pp. 47-59, 1966. · Zbl 0134.26903 · doi:10.2140/pjm.1966.16.47
[18] K. McCrimmon, “Macdonald/s theorem with inverses,” Pacific Journal of Mathematics, vol. 21, pp. 315-325, 1967. · Zbl 0166.04001 · doi:10.2140/pjm.1967.21.315
[19] H. Upmeier, Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics, vol. 67 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1987. · Zbl 0608.17013
[20] H. Hanche-Olsen and E. Størmer, Jordan Operator Algebras, vol. 2 of Monographs and Studies in Mathematics, Pitman, Boston, Mass, USA, 1984. · Zbl 0561.46031
[21] H. Choda, “An extremal property of the polar decomposition in von Neumann algebras,” Proceedings of the Japan Academy, vol. 46, pp. 341-344, 1970. · Zbl 0207.44402 · doi:10.3792/pja/1195520348
[22] J. D. M. Wright and M. A. Youngson, “A Russo-Dye theorem for Jordan C\ast -algebras,” in Functional Analysis: Surveys and Recent Results (Proceedings of the Conference Held at Paderborn, 1976), vol. 27 of North-Holland Math. Studies, pp. 279-282, North Holland, Amsterdam, The Netherlands, 1977. · Zbl 0372.46060
[23] W. Kaup and H. Upmeier, “Jordan algebras and symmetric Siegel domains in Banach spaces,” Mathematische Zeitschrift, vol. 157, no. 2, pp. 179-200, 1977. · Zbl 0357.32018 · doi:10.1007/BF01215150 · eudml:172606
[24] R. Braun, W. Kaup, and H. Upmeier, “A holomorphic characterization of Jordan C\ast -algebras,” Mathematische Zeitschrift, vol. 161, no. 3, pp. 277-290, 1978. · Zbl 0385.32002 · doi:10.1007/BF01214510 · eudml:172707
[25] J. D. M. Wright and M. A. Youngson, “On isometries of Jordan algebras,” Journal of the London Mathematical Society. Second Series, vol. 17, no. 2, pp. 339-344, 1978. · Zbl 0384.46041 · doi:10.1112/jlms/s2-17.2.339
[26] L. A. Harris, “Bounded symmetric homogeneous domains in infinite dimensional spaces,” in Proceedings on Infinite Dimensional Holomorphy (Proceedings of an International Conference, University of Kentucky, Lexington, Kentucky, 1973), vol. 364 of Lecture Notes in Math., pp. 13-40, Springer, Berlin, Germany, 1974. · Zbl 0293.46049
[27] R. Berntzen, “Convex combinations of extreme points of the closed unit ball in AW\ast -algebras,” in preparation. · Zbl 0896.46038
[28] R. R. Phelps, “Extreme points in function algebras,” Duke Mathematical Journal, vol. 32, no. 2, pp. 267-277, 1965. · Zbl 0139.07401 · doi:10.1215/S0012-7094-65-03226-6
[29] A. A. Siddiqui, “Positivity of invertibles in unitary isotopes of JB\ast -algebras,” in preparation.