Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A characterization of inner ideals in $ {\rm JB}\sp *$-triples


Authors: C. M. Edwards and G. T. Rüttimann
Journal: Proc. Amer. Math. Soc. 116 (1992), 1049-1057
MSC: Primary 46L70
DOI: https://doi.org/10.1090/S0002-9939-1992-1102856-1
MathSciNet review: 1102856
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that a norm-closed subtriple $ B$ of a $ J{B^ * }$-triple $ A$ is an inner ideal if and only if every bounded linear functional on $ B$ has a unique norm-preserving extension to a bounded linear functional on $ A$. It follows that the norm-closed subtriples $ B$ of a $ {C^ * }$-algebra $ A$ that enjoy this unique extension property are precisely those of the form $ e{A^{ * * }}f \cap A$ where $ (e,f)$ is a pair of centrally equivalent open projections in the $ {W^ * }$-envelope $ {A^{ * * }}$ of $ A$.


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

  • [1] E. M. Alfsen and F. W. Shultz, Non-commutative spectral theory for affine function spaces on convex sets, Mem. Amer. Math. Soc., vol. 172, Amer. Math. Soc., Providence, RI, 1976. MR 0412822 (54:943)
  • [2] E. M. Alfsen, F. W. Shultz, and E. Størmer, A Gelfand-Neumark theorem for Jordan algebras, Adv. Math. 28 (1978), 11-56. MR 0482210 (58:2292)
  • [3] T. J. Barton and R. M. Timoney, Weak*-continuity of Jordan triple products and its applications, Math. Scand. 59 (1986), 177-191. MR 884654 (88d:46129)
  • [4] T. J. Barton, T. Dang, and G. Horn, Normal representations of Banach Jordan triple systems, Proc. Amer. Math. Soc. 102 (1987), 551-555. MR 928978 (89c:46065)
  • [5] M. Battaglia, Order theoretic type decomposition of JBW*-triples, Quart. J. Math. Oxford Ser. 2 42 (1991), 129-147. MR 1107278 (92d:46171)
  • [6] S. Dineen, Complete holomorphic vector fields in the second dual of a Banach space, Math. Scand. 59 (1986), 131-142. MR 873493 (88h:32029)
  • [7] C. M. Edwards, Ideal theory in JB-algebras, J. London Math. Soc. (2) 16 (1977), 507-513. MR 0487459 (58:7092)
  • [8] -, On the facial structure of a JB-algebra, J. London Math. Soc. (2) 19 (1979), 335-344. MR 533334 (80e:46033)
  • [9] -, On Jordan $ {W^ * }$-algebras, Bull. Sci. Math. (2) 104 (1980), 393-403. MR 602407 (82g:46097)
  • [10] C. M. Edwards and G. T. Rüttimann, On the facial structure of the unit balls in a JBW*-triple and its predual, J. London Math. Soc. (2) 38 (1988), 317-332. MR 966303 (90b:46129)
  • [11] -, Inner ideals in $ {W^ * }$-algebras, Michigan Math. J. 36 (1989), 147-159.
  • [12] -, On inner ideals in ternary algebras, Math. Z. 204 (1990), 309-318. MR 1107465 (92k:46119)
  • [13] -, Inner ideals in $ {C^ * }$-algebras, Math. Ann. 290 (1991), 621-628. MR 1119941 (93b:46111)
  • [14] C. M. Edwards, G. T. Rüttimann, and S. Yu. Vasilovsky, Inner ideals in exceptional JBW*-triples, Michigan Math J. (to appear). MR 1214059 (94d:46068)
  • [15] -, Invariant inner ideals in $ {W^ * }$-algebras, preprint.
  • [16] Y. Friedman and B. Russo, Structure of the predual of a JBW*-triple, J. Reine Angew. Math. 356 (1985), 67-89. MR 779376 (86f:46073)
  • [17] H. Hanche-Olsen and E. Størmer, Jordan operator algebras, Pitman, London, 1984.
  • [18] G. Horn, Characterization of the predual and the ideal structure of a JBW*-triple, Math. Scand. 61 (1987), 117-133. MR 929400 (89a:46140)
  • [19] G. Horn and E. Neher, Classification of continuous JBW*-triples, Trans. Amer. Math. Soc. 306 (1988), 553-578. MR 933306 (89c:46090)
  • [20] W. Kaup, Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), 503-529. MR 710768 (85c:46040)
  • [21] M. Kusuda, Unique state extension and hereditary $ {C^ * }$-algebras, Math. Ann. 288 (1990), 201-209. MR 1075765 (92e:46122)
  • [22] E. Neher, Jordan triple systems by the grid approach, Lecture Notes in Math., vol. 1280, Springer-Verlag, Berlin, Heidelberg, and New York, 1987. MR 911879 (89b:17024)
  • [23] -, Jordan pairs with finite grids, Comm. Algebra 19 (1991), 455-478. MR 1100357 (92g:17048)
  • [24] G. K. Pedersen, $ {C^ * }$-algebras and their automorphism groups, Academic Press, London, 1979. MR 548006 (81e:46037)
  • [25] R. R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximations, Trans. Amer. Math. Soc. 95 (1960), 238-255. MR 0113125 (22:3964)
  • [26] F. W. Shultz, On normed Jordan algebras which are Banach dual spaces, J. Funct. Anal. 31 (1979), 360-376. MR 531138 (80h:46096)
  • [27] H. Upmeier, Symmetric Banach manifolds and Jordan $ {C^ * }$-algebra, North-Holland, Amsterdam, 1985. MR 776786 (87a:58022)
  • [28] J. D. M. Wright, Jordan $ {C^ * }$-algebras, Michigan Math. J. 24 (1977), 291-302. MR 0487478 (58:7108)
  • [29] M. A. Youngson, A Vidav theorem for Banach Jordan algebras, Math. Proc. Cambridge Philos. Soc. 84 (1978), 263-272. MR 0493372 (58:12397)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46L70

Retrieve articles in all journals with MSC: 46L70


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1102856-1
Keywords: $ J{B^ * }$-triple, inner ideal
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society