×

Weakly compact operators on Jordan triples. (English) Zbl 0627.46061

We obtain some criteria for the weak compactness of bounded operators on \(JB^ *\)-triples, extending the results of Akemann et al. and Jarchow for \(C^ *\)-algebras. From this we derive some Banach space properties for \(JB^ *\)-triples.

MSC:

46H70 Nonassociative topological algebras
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
17C65 Jordan structures on Banach spaces and algebras
46L70 Nonassociative selfadjoint operator algebras
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Akemann, C.A., Dodds, P.G., Gamlen, J.L.B.: Weak compactness in the dual space of aC *-algebra. J. Funct. Anal.10, 446-450 (1972) · Zbl 0238.46058
[2] Alvermann, K., Janssen, G.: Real and complex noncommutative Jordan Banach algebras. Math. Z.185, 105-113 (1984) · Zbl 0525.46038
[3] Barton, T., Friedman, Y.: Grothendieck’s inequality forJB *-triples and applications. J. London Math. Soc.36, 513-523 (1987) · Zbl 0652.46039
[4] Beauzamy, B.: Opérateurs uniformément convexifiants. Stud. Math.57, 103-139 (1976) · Zbl 0372.46016
[5] Chu, C.-H., Iochum, B.: On the Radon-Nikodým property in Jordan triples. Proc. Am. Math. Soc.99, 462-464 (1987) · Zbl 0617.46026
[6] Davis, W.J., Figiel, T., Johnson, W.B., Pelczynski, A.: Factoring weakly compact operators. J. Funct. Anal.17, 311-327 (1974) · Zbl 0306.46020
[7] Dineen, S.: complete holomorphic vector fields on the second dual of a Banach space. Math. Scand.59, 131-142 (1986) · Zbl 0625.46055
[8] Enflo, P.: Banach spaces which can be given an equivalent uniformly convex norm. Isr. J. Math.13, 281-288 (1973) · Zbl 0259.46012
[9] Friedman, Y., Russo, B.: The Gelfand-Naimark theorem forJB *-triples. Duke Math. J.53, 139-148 (1986) · Zbl 0637.46049
[10] Hanche-Olsen, H., Stormer, E.: Jordan Operator algebras. London: Pitman 1984
[11] Harris, C.: A generalization ofC *-algebras. Proc. Lond. Math. Soc.42, 331-361 (1981) · Zbl 0476.46054
[12] Horn, G.: Klassifikation derJBW *-tripel von typ I. Dissertation, Tübingen (1984)
[13] Horn, G., Neher, E.: Classification of continuousJBW *-triples. Trans. Am. Math. Soc. (in press) · Zbl 0659.46063
[14] Iochum, B.: Cônes autopolaires et algébres de Jordan. Lecture Notes in Math. 1049. Berlin Heidelberg New York: Springer 1984 · Zbl 0556.46040
[15] Janssen, G.: Factor representations of type I for noncommutativeJB andJB *-algebras. Preprint (1983)
[16] Jarchow, H.: Weakly compact operators onC(K) and onC *-algebras. Lecture Notes, Ecole d’Automne CIMPA, Nice (1986)
[17] Jarchow, H.: On weakly compact operators onC *-algebras. Math. Ann.273, 341-343 (1986) · Zbl 0593.47016
[18] Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z.183, 503-529 (1983) · Zbl 0519.32024
[19] Pedersen, G.K.:C *-algebras and their automorphism groups. London: Academic Press 1979 · Zbl 0416.46043
[20] Pietsch, A.: Operator ideals. Amsterdam: North-Holland 1980 · Zbl 0434.47030
[21] Pisier, G.:K-convexity, Proc. Res. Workshop on Banach space theory. University of Iowa Press (1982)
[22] Rosenthal, H.P.: On subspaces ofL P . Ann. Math.97, 344-373 (1973) · Zbl 0253.46049
[23] Takesaki, M.: Theory of operator algebras I. Berlin Heidelberg New York: Springer 1979 · Zbl 0436.46043
[24] Upmeier, H.: Symmetric Banach manifolds and JordanC *-algebras. Amsterdam: North-Holland 1985 · Zbl 0561.46032
[25] Wright, J.D.M.: JordanC *-algebras. Mich. Math. J.24, 291-302 (1977) · Zbl 0384.46040
[26] Spain, P.G.: A generalization of a theorem of Grothendieck. Quant. J. Math. Oxford27, 475-479 (1976) · Zbl 0341.46007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.