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Almost transitivity of some function spaces

Published online by Cambridge University Press:  24 October 2008

Peter Greim ,
James E. Jamison  and
Anna Kamińska
Peter Greim
Affiliation:
Department of Mathematics, The Citadel, Charleston, SC 29409, U.S.A.
James E. Jamison
Affiliation:
Department of Mathematics, The University of Memphis, Memphis, TN 38152, U.S.A.
Anna Kamińska
Affiliation:
Department of Mathematics, The University of Memphis, Memphis, TN 38152, U.S.A.
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Abstract

The almost transitive norm problem is studied for Lp (μ, X), C(K, X) and for certain Orlicz and Musielak-Orlicz spaces. For example if p ≠ 2 < ∞ then Lp (μ) has almost transitive norm if and only if the measure μ is homogeneous. It is shown that the only Musielak-Orlicz space with almost transitive norm is the Lp-space. Furthermore, an Orlicz space has an almost transitive norm if and only if the norm is maximal. Lp (μ, X) has almost transitive norm if Lp(μ) and X have. Separable spaces with non-trivial Lp-structure fail to have transitive norms. Spaces with nontrivial centralizers and extreme points in the unit ball also fail to have almost transitive norms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 3 , November 1994 , pp. 475 - 488
Copyright
Copyright © Cambridge Philosophical Society 1994

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References

REFERENCES

[1]Banach, S.. Theory of Linear Operations (North-Holland, 1987).Google Scholar
[2]Behrends, E.. L p-Struktur in Banachraumen. Studia Math. 55 (1976), 71–85.CrossRefGoogle Scholar
[3]Behrends, E. et al. L p-Structure in Real Banach Spaces. Lecture Notes in Mathematics 613 (Springer-Verlag, 1977).Google Scholar
[4]Behrends, E.. M-Structure and the Banach-Stone Theorem. Lecture Notes in Mathematics 736 (Springer-Verlag, 1979).Google Scholar
[5]Behrends, E.. Isomorphic Banach-Stone theorems and isomorphisms which are close to isometries. Pacific J. Math. 133, no. 2 (1988), 229–250.Google Scholar
[6]Cambern, M. and Jarosz, K.. Ultraproducts, ε-multipliers, and isomorphisms. Proc. Amer. Math. Soc. 105 (1989), 929–937.Google Scholar
[7]Cowie, R. E.. A note on uniquely maximal Banach spaces. Proc. Edinburgh Math. Soc. 26 (1983), 85–87.Google Scholar
[8]Fleming, R. G. and Jamison, J. E.. Isometries of Banach spaces, a survey in The Riemann Legacy Volume (Hadronic Press), (to appear).Google Scholar
[9]Greim, P.. The centralizer of Bochner L ∞-spaces. Math. Ann. 260 (1982), 463–468.CrossRefGoogle Scholar
[10]Greim, P.. Hilbert spaces have the Banach-Stone property for Bochner spaces. Bull. Austr. Math. Soc. 27 (1983), 121–128.CrossRefGoogle Scholar
[11]Greim, P.. Isometries and L p-structure of separably valued Bochner L p-spaces, in Measure theory and its applications. Lecture Notes in Mathematics 1033 (Springer-Verlag, 1983).Google Scholar
[12]Greim, P.. Banach spaces with the L 1-Banach-Stone property. Trans. Amer. Math. Soc. 287 (1985), 819–828.Google Scholar
[13]Greim, P. and Jamison, J. E.. Hilbert spaces have the strong Banach-Stone property for Bochner spaces. Math. Z. 196 (1987), 511–515.CrossRefGoogle Scholar
[14]Jamison, J. E., Kamińska, A. and Lin, Pei-Kee. Isometries of Musielak-Orlicz spaces II, Studia Math. 104 (1) (1993), 75–89.Google Scholar
[15]Kalton, N. J. and Radrianantoanina, B.. Surjective isometries on rearrangement-invariant spaces, Quart. J. Math. Oxford (to appear).Google Scholar
[16]Koppelberg, S.. Handbook of Boolean Algebras, vol. I (North-Holland, 1989).Google Scholar
[17]Krasnoselskiiand, M. A. and Rutickii, Ya. B.. Convex Functions and Orlicz Spaces (Noordhoff Ltd. 1961).Google Scholar
[18]Elton Lacy, H.. The Isometric Theory of Classical Banach Spaces (Springer-Verlag, 1974).Google Scholar
[19]Lamperti, J.. On the isometries of certain function spaces. Pacific J. Math. 8 (1958), 459–466.Google Scholar
[20]Lin, Pei-Kee. The isometries of L 2 (Ω, X). Ill. J. Math. 33 no. 4 (1989), 621–630.Google Scholar
[21]Lindenstrauss, J. and Tzafriri, L.. Classical Banach spaces II (Springer-Verlag, 1979).Google Scholar
[22]Musielak, J.. Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics 1034 (Springer-Verlag, 1983).Google Scholar
[23]Rolewicz, S.. Metric Linear Spaces. (Polish Scientific Publishers and D. Reidel Publishing Company, 1984).Google Scholar
[24]Royden, H. L.. Real Analysis (Macmillan, 1988).Google Scholar
[25]Sims, B.. Ultra-techniques in Banach space theory. Queens Papers in Pure and Applied Mathematics, no. 60 (Queen's University, Kingston, Ontario, Canada, 1982).Google Scholar
[26]Sourour, A. R.. The isometries of L p(Ω, X). J. Funct. Anal. 30 (1978), 276–285.Google Scholar
[27]Wood, G. V.. Maximal symmetry in Banach spaces. Proc. Royal Irish Acad. 82A, no. 2 (1982), 177–186.Google Scholar
[28]Zaidenberg, M. G.. Groups of isometries of Orlicz spaces. Soviet Math. Dokl. 17 (1976), 432–436.Google Scholar