Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T08:29:41.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Compact tripotents in bi-dual JB*-triples1

Published online by Cambridge University Press:  24 October 2008

C. Martin Edwards  and
Gottfried T. Rüttimann
C. Martin Edwards
Affiliation:
The Queen's College, Oxford
Gottfried T. Rüttimann
Affiliation:
University of Berne, Berne, Switzerland
Get access Rights & Permissions [Opens in a new window]

Abstract

The set consisting of the partially ordered set of tripotents in a JBW*-triple C with a greatest element adjoined forms a complete lattice. This paper is mainly concerned with the situation in which C is the second dual A** of a complex Banach space A and, more particularly, when A is itself a JB*-triple. A subset of consisting of the set of tripotents compact relative to A (denned in Section 4) with a greatest element adjoined is studied. It is shown to be an atomic complete lattice with the properties that the infimum of an arbitrary family of elements of is the same whether taken in or in and that every decreasing net of non-zero elements of has a non-zero infimum. The relationship between the complete lattice and the complete lattice where B is a Banach space such that B** is a weak*-closed subtriple of A** is also investigated. When applied to the special case in which A is a C*-algebra the results provide information about the set of compact partial isometries relative to A and are closely related to those recently obtained by Akemann and Pedersen. In particular it is shown that a partial isometry is compact relative to A if and only if, in their terminology, it belongs locally to A. The main results are applied to this and other examples.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 1 , July 1996 , pp. 155 - 173
Copyright
Copyright © Cambridge Philosophical Society 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Akemann, C. A. and Pedersen, G. K.. Facial structure in operator algebra theory. Proc. London Math. Soc. 64 (1992), 418–448.Google Scholar
[2]Arazy, J. and Solel, B.. Isometrics of non-self-adjoint operator algebras. J. Functional Anal. 90 (1990), 284–305.Google Scholar
[3]Barton, T. J. and Timoney, R. M.. Weak* continuity of Jordan triple products and its applications. Math. Scand. 59 (1986), 177–191.CrossRefGoogle Scholar
[4]Barton, T. J., Dang, T. and Hoen, G.. Normal representations of Banach Jordan triple systems. Proc. Amer. Math. Soc. 102 (1987), 551–555.Google Scholar
[5]Battaglia, M.. Order theoretic type decomposition of JBW*-triples. Quart. J. Math. Oxford 42 (1991), 129–147.CrossRefGoogle Scholar
[6]Dineen, S.. The second dual of a JB*-triple system; in Complex analysis, functional analysis and approximation theory (North Holland, 1986).Google Scholar
[7]Edwards, C. M.. On Jordan W*-algebras. Bull. Sc. Math. 2e serie 104 (1980), 393–403.Google Scholar
[8]Edwards, C. M. and Rüttimann, G. T.. On the facial structure of the unit balls in a JBW*-triple and its predual. J. London Math. Soc. 38 (1988), 317–332.Google Scholar
[9]Edwards, C. M. and Rüttimann, G. T.. Inner ideals in W*-algebras. Michigan Math. J. 36 (1989), 147–159.CrossRefGoogle Scholar
[10]Edwards, C. M. and Rüttimann, G. T.. Inner ideals in C*-algebras. Math. Ann. 290 (1991), 621–628.Google Scholar
[11]Edwards, C. M. and Rüttimann, G. T.. A characterization of inner ideals in JB*-triples. Proc. Amer. Math. Soc. 116 (1992), 1049–1057.Google Scholar
[12]Edwards, C. M., Rüttimann, G. T. and Vasilovsky, S. Yu.. Inner ideals in exceptional JBW*-triples. Michigan Math. J. 40 (1993), 139–152.CrossRefGoogle Scholar
[13]Edwards, C. M. and Rüttimann, G. T.. Structural projections on JBW*-triples. J. London Math. Soc. (2) 53 (1996), 354–368.Google Scholar
[14]Edwards, C. M., McCrimmon, K. and Rüttimann, G. T.. The range of a structural projection J. Functional Anal, (to appear).Google Scholar
[15]Friedman, Y. and Russo, B.. Structure of a predual of a JBW*-triple. J. reine Angew. Math. 356 (1985), 67–89.Google Scholar
[16]Friedman, Y. and Russo, B.. The Gelfand-Naimark theorem for JB*-triples. Duke Math. J. 53 (1986), 139–148.CrossRefGoogle Scholar
[17]Friedman, Y. and Russo, B.. Conditional expectation and bicontractive projections on Jordan C*-algebras and their generalizations. Math. Z. 194 (1987), 227–236.Google Scholar
[18]Hanche-Olsen, H. and Størmer, E.. Jordan operator algebras. (Pitman, 1984).Google Scholar
[19]Horn, G.. Characterization of the predual and the ideal structure of a JBW*-triple. Math. Scand. 61 (1987), 117–133.Google Scholar
[20]Jacobson, N., Structure and representation of Jordan algebras (American Mathematical Society, 1968).CrossRefGoogle Scholar
[21]Kaup, W.. Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183 (1983), 503–529.CrossRefGoogle Scholar
[22]Kaup, W.. Contractive projections on Jordan C*-algebras and generalizations. Math. Scand. 54 (1984), 95–100.CrossRefGoogle Scholar
[23]Kaup, W. and Upmeier, H.. Banach spaces with biholomorphically equivalent unit balls are isomorphic. Proc. Amer. Math. Soc. 58 (1976), 129–133.Google Scholar
[24]Koecher, M.. An elementary approach to bounded symmetric domains. Rice University Lecture Notes (Houston, 1969).Google Scholar
[25]Loos, O.. Jordan pairs. Lecture Notes in Mathematics 460 (Springer-Verlag, 1975).CrossRefGoogle Scholar
[26]Meyberg, K.. Jordan-Tripelsysteme und die Koecher-Konstruktion von Lie-Algebren. Math. Z. 115 (1970), 58–78.CrossRefGoogle Scholar
[27]Neher, E.. Jordan triple systems by the grid approach. Lecture Notes in Mathematics 1280 (Springer, 1987).CrossRefGoogle Scholar
[28]Pedersen, G. K.. C*-algebras and their automorphism groups. London Mathematical Society Monographs 14. (Academic Press, 1979).Google Scholar
[29]Sakai, S.. C*-algebras and W*-algebras (Springer, 1971).Google Scholar
[30]Upmeier, H.. Symmetric Banach manifolds and Jordan C*-algebras (North Holland, 1985).Google Scholar
[31]Upmeier, H.. Jordan algebras in analysis, operator theory, and quantum mechanics (American Mathematical Society, 1986).Google Scholar
[32]Vigué, J. P.. Le groupe des automorphismes analytiques d'un domaine borné d'un espace de Banach complexe. Applications aux domaines bornés symmétriques. Ann. Sci. Éc. Norm. Sup. 4e série 9 (1976), 203–282.CrossRefGoogle Scholar
[33]Vigué, J. P.. Les domaines bornés symmétriques d'un espace de Banach complexe et les systemes triples de Jordan. Math. Ann. 229 (1977), 223–231.CrossRefGoogle Scholar
[34]Vigué, J. P.. Automorphismes analytiques des produits de domaines bornes. Ann. Sci. Éc. Norm. Sup. 4esérie 11 (1978), 229–246.Google Scholar
[35]Vigué, J. P.. Les automorphismes analytiques isométriques d'une variété complexe normeé. Bull. Soc. Math. France 110 (1982), 49–73.CrossRefGoogle Scholar
[36]Wright, J. D. M.. Jordan C*-algebras. Michigan Math. J. 24 (1977), 291–302.CrossRefGoogle Scholar
[37]Youngson, M. A.. A Vidav theorem for Banach Jordan algebras. Math. Proc. Camb. Phil. Soc. 84 (1978), 263–272.Google Scholar