Means on C V p ( G ) -subspaces of C V p ( G ) with RNP and Schur property

Françoise Lust-Piquard

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 4, page 969-1006
  • ISSN: 0373-0956

Abstract

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Let G be a locally compact abelian group and C V p ( G ) ( 1 p 2 ) be the space of bounded convolution operators: L p ( G ) L p ( G ) . We generalize to C V p ( G ) some results which are well known for C V 2 ( G ) (or rather for L ( G ^ ) ): we define and study “invariant means” on C V p ( G ) , and we show that if E G is compact and scattered the space C V p ( E ) (convolution operators which are supported on E ) has the Schur property and is the norm closure of finitely supported measures. We also give some consequences of these results.

How to cite

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Lust-Piquard, Françoise. "Means on $CV_p(G)$-subspaces of $CV_p(G)$ with RNP and Schur property." Annales de l'institut Fourier 39.4 (1989): 969-1006. <http://eudml.org/doc/74864>.

@article{Lust1989,
abstract = {Let $G$ be a locally compact abelian group and $CV_ p(G)$$(1\le p\le 2)$ be the space of bounded convolution operators: $L^ p(G)\rightarrow L^ p(G)$. We generalize to $CV_ p(G)$ some results which are well known for $CV_2(G)$ (or rather for $L^\{\infty \}(\hat\{G\})$): we define and study “invariant means” on $CV_ p(G)$, and we show that if $E\subset G$ is compact and scattered the space $CV_ p(E)$ (convolution operators which are supported on $E$) has the Schur property and is the norm closure of finitely supported measures. We also give some consequences of these results.},
author = {Lust-Piquard, Françoise},
journal = {Annales de l'institut Fourier},
keywords = {Radon-Nikodym property; locally compact abelian group; space of bounded convolution operators; invariant means; Schur property; finitely supported measures},
language = {eng},
number = {4},
pages = {969-1006},
publisher = {Association des Annales de l'Institut Fourier},
title = {Means on $CV_p(G)$-subspaces of $CV_p(G)$ with RNP and Schur property},
url = {http://eudml.org/doc/74864},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Lust-Piquard, Françoise
TI - Means on $CV_p(G)$-subspaces of $CV_p(G)$ with RNP and Schur property
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 4
SP - 969
EP - 1006
AB - Let $G$ be a locally compact abelian group and $CV_ p(G)$$(1\le p\le 2)$ be the space of bounded convolution operators: $L^ p(G)\rightarrow L^ p(G)$. We generalize to $CV_ p(G)$ some results which are well known for $CV_2(G)$ (or rather for $L^{\infty }(\hat{G})$): we define and study “invariant means” on $CV_ p(G)$, and we show that if $E\subset G$ is compact and scattered the space $CV_ p(E)$ (convolution operators which are supported on $E$) has the Schur property and is the norm closure of finitely supported measures. We also give some consequences of these results.
LA - eng
KW - Radon-Nikodym property; locally compact abelian group; space of bounded convolution operators; invariant means; Schur property; finitely supported measures
UR - http://eudml.org/doc/74864
ER -

References

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  1. [BJM] J. F. BERGLUND, H. D. JUNGHEN and P. MILNES, Compact right topological semi-groups and generalization of almost periodicity, Lecture Notes in Maths., n° 663, 1978, Springer-Verlag. Zbl0406.22005
  2. [BD] F. F. BONSALL and J. DUNCAN, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Math. Soc. Lecture Notes series, 2 (1971) and 10 (1973). Zbl0262.47001
  3. [Eb1] W. F. EBERLEIN, Abstract ergodic theorems and weak almost periodic functions, Trans. of the AMS, vol. 67 (1949), 217-240. Zbl0034.06404MR12,112a
  4. [Eb2] W. F. EBERLEIN, The point spectrum of weakly almost periodic functions, Mich. Math. J., vol. 3 (1955), 137-139. Zbl0073.30702MR18,583b
  5. [Ey] P. EYMARD, Algèbres Ap et convoluteurs de Lp, Séminaire Bourbaki, exposé 367, nov. 1969, pp. 1-18. Zbl0264.43006
  6. [G1] P. GLOWACKI, A note on functions with scattered spectra on lca groups, Studia Math., vol. 70 (1981), 147-152. Zbl0489.43001MR83c:43008
  7. [Gra] E. GRANIRER, On some spaces of linear functionals on the algebras Ap(G) for locally compact groups, Colloquium Math., vol. LII (1987), 119-132. Zbl0649.43004MR88k:43006
  8. [Gre] F. P. GREENLEAF, Invariant means on topological groups, Van Nostrand, 1969. Zbl0174.19001MR40 #4776
  9. [H1] C. HERZ, Harmonic synthesis for subgroups, Annales de l'Institut Fourier, vol. 23-3 (1973), 91-123. Zbl0257.43007MR50 #7956
  10. [H2] C. HERZ, Une généralisation de la notion de transformée de Fourier Stieltjes, Annales de l'Institut Fourier, vol. 24-3 (1974), 145-157. Zbl0287.43006MR54 #13466
  11. [Loh1] N. LOHOUÉ, Algèbres Ap(G) et convoluteurs de Lp(G), Thèse Université Paris-Sud-Orsay (1971). 
  12. [Loh2] N. LOHOUÉ, Approximation et transfert d'opérateurs de convolution, Annales de l'Institut Fourier, vol. 26-4 (1976), 133-150. Zbl0331.43008MR57 #7036
  13. [Loo] L. H. LOOMIS, The spectral characterization of a class of almost periodic functions, Annals of Math., vol. 72, n° 2 (1960), 362-368. Zbl0094.05801MR22 #11255
  14. [L-P1] F. LUST-PIQUARD, L'espace des fonctions presque périodiques dont le spectre est contenu dans un ensemble compact dénombrable a la propriété de Schur, Colloquium Math., vol. XLI (1979), 273-284. Zbl0462.43007MR81m:43011
  15. [L-P2] F. LUST-PIQUARD, Éléments ergodiques et totalement ergodiques dans L∞(Γ), Studia Math., vol. LXIX (1981), 191-225. Zbl0476.43001MR84h:43003
  16. [L-P3] F. LUST-PIQUARD, Propriétés géométriques des sous-espaces invariants par translation de C(G) et L1(G). Séminaire sur la géométrie des espaces de Banach, École Polytechnique (1977-1978), exposé 26. Zbl0386.46020
  17. [L-P4] F. LUST-PIQUARD, Produits tensoriels projectifs d'espaces de Banach faiblement séquentiellement complets, Colloquium Math., vol. 36 (1976), 255-267. Zbl0356.46058MR55 #11072
  18. [P] J. P. PIER, Amenable locally compact groups, Wiley Interscience, 1984. Zbl0621.43001MR86a:43001
  19. [R] H. P. ROSENTHAL, A characterization of Banach spaces containing l1, Proc. Nat. Acad. Sci. USA, vol. 71 (1974), 2411-2413. Zbl0297.46013MR50 #10773
  20. [S] W. SCHACHERMAYER, Some translation invariant subspaces of C(G) which have the strong Schur property, Groupe de travail sur les espaces invariants par translation, Publications mathématiques d'Orsay, n° 89-02 (1989). Zbl0704.43003MR90m:43007
  21. [V] N. Th. VAROPOULOS, Tensor algebras and harmonic analysis, Acta Math., vol. 119 (1967), 51-112. Zbl0163.37002MR39 #1911
  22. [W1] G. S. WOODWARD, Invariant means and ergodic sets in Fourier analysis, Pacific J. of Maths, vol. 54-2 (1974), 281-299. Zbl0307.43006MR51 #11021
  23. [W2] G. S. WOODWARD, The generalized almost periodic part of an ergodic function, Studia Math., vol. 50 (1974), 103-116. Zbl0283.42019MR52 #3889

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