Relative weak compactness of orbits in Banach spaces associated with locally compact groups
HTML articles powered by AMS MathViewer
- by Colin C. Graham and Anthony T. M. Lau PDF
- Trans. Amer. Math. Soc. 359 (2007), 1129-1160 Request permission
Abstract:
We study analogues of weak almost periodicity in Banach spaces on locally compact groups. i) If $\mu$ is a continous measure on the locally compact abelian group $G$ and $f\in L^\infty (\mu )$, then $\{\gamma f:\gamma \in \widehat G\}$ is not relatively weakly compact. ii) If $G$ is a discrete abelian group and $f\in \ell ^\infty (G)\backslash C_o(G)$, then $\{\gamma f:\gamma \in E\}$ is not relatively weakly compact if $E\subset \widehat G$ has non-empty interior. That result will follow from an existence theorem for $I_o$-sets, as follows. iii) Every infinite subset of a discrete abelian group $\Gamma$ contains an infinite $I_o$-set such that for every neighbourhood $U$ of the identity of $\widehat \Gamma$ the interpolation (except at a finite subset depending on $U$) can be done using at most 4 point masses. iv) A new proof that $B(G)\subset WAP(G)$ for abelian groups is given that identifies the weak limits of translates of Fourier-Stieltjes transforms. v) Analogous results for $C_o(G)$, $A_p(G)$, and $M_p(G)$ are given. vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of $\widehat G$ in $L^\infty (\mu )$, for abelian $G$; and when $\rho$ is a continuous homomorphism of the locally compact group $\Gamma$ into the unitary elements of a von Neumann algebra $\mathcal {M}$, the weak* closure of $\rho (\Gamma )$ is studied.References
- Jesús Gil de Lamadrid and Loren N. Argabright, Almost periodic measures, Mem. Amer. Math. Soc. 85 (1990), no. 428, vi+219. MR 979431, DOI 10.1090/memo/0428
- John F. Berglund, Hugo D. Junghenn, and Paul Milnes, Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. MR 999922
- Gavin Brown, Riesz products and generalized characters, Proc. London Math. Soc. (3) 30 (1975), 209–238. MR 372530, DOI 10.1112/plms/s3-30.2.209
- Gavin Brown, Colin Graham, and William Moran, Translation and symmetry in $M(G)$, Symposia Mathematica, Vol. XXII (Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976) Academic Press, London, 1977, pp. 371–392. MR 0487272
- R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- Charles F. Dunkl and Donald E. Ramirez, Sections induced from weakly sequentially complete spaces, Studia Math. 49 (1973), 95–97. MR 333591, DOI 10.4064/sm-49-1-95-97
- Myriam Déchamps-Gondim, Ensembles de Sidon topologiques, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 51–79 (French, with English summary). MR 340981, DOI 10.5802/aif.424
- W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240. MR 36455, DOI 10.1090/S0002-9947-1949-0036455-9
- W. F. Eberlein, A note on Fourier-Stieltjes transforms, Proc. Amer. Math. Soc. 6 (1955), 310–312. MR 68030, DOI 10.1090/S0002-9939-1955-0068030-2
- R. E. Edwards and G. I. Gaudry, Littlewood-Paley and multiplier theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90, Springer-Verlag, Berlin-New York, 1977. MR 0618663, DOI 10.1007/978-3-642-66366-6
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628, DOI 10.24033/bsmf.1607
- Jorge Galindo and Salvador Hernández, The concept of boundedness and the Bohr compactification of a MAP abelian group, Fund. Math. 159 (1999), no. 3, 195–218. MR 1680642, DOI 10.4064/fm-159-3-195-218
- Colin C. Graham, Arens regularity for quotients $A_p(E)$ of the Herz algebra, Bull. London Math. Soc. 34 (2002), no. 4, 457–468. MR 1897425, DOI 10.1112/S0024609302001121
- Colin C. Graham and O. Carruth McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 238, Springer-Verlag, New York-Berlin, 1979. MR 550606, DOI 10.1007/978-1-4612-9976-9
- Kathryn E. Hare and L. Thomas Ramsey, $I_0$ sets in non-abelian groups, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 1, 81–98. MR 1990833, DOI 10.1017/S0305004103006790
- S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, Colloq. Math. 12 (1964), 23–39. MR 167785, DOI 10.4064/cm-12-1-23-39
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- Carl Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123 (English, with French summary). MR 355482, DOI 10.5802/aif.473
- Carl Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 3, xiii, 145–157 (French, with English summary). MR 425511
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 0188387
- B. Host, Le théorème des idempotents dans $B(G)$, Bull. Soc. Math. France 114 (1986), no. 2, 215–223 (French, with English summary). MR 860817, DOI 10.24033/bsmf.2055
- B. Host, J.-F. Méla, and F. Parreau, Analyse harmonique des mesures, Astérisque 135-136 (1986), 261 (French). MR 839692
- Bernard Host, Jean-François Méla, and François Parreau, Nonsingular transformations and spectral analysis of measures, Bull. Soc. Math. France 119 (1991), no. 1, 33–90 (English, with French summary). MR 1101939, DOI 10.24033/bsmf.2157
- Kenneth Kunen and Walter Rudin, Lacunarity and the Bohr topology, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 1, 117–137. MR 1681658, DOI 10.1017/S030500419800317X
- Anthony To Ming Lau and James C. S. Wong, Weakly almost periodic elements in $L_\infty (G)$ of a locally compact group, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1031–1036. MR 991701, DOI 10.1090/S0002-9939-1989-0991701-0
- Jorge M. López and Kenneth A. Ross, Sidon sets, Lecture Notes in Pure and Applied Mathematics, Vol. 13, Marcel Dekker, Inc., New York, 1975. MR 0440298
- Jean-François Méla, Suites lacunaires de Sidon, ensembles propres et points exceptionnels, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 2, 533–538 (French). MR 178308, DOI 10.5802/aif.189
- Jean-François Méla, Sur certains ensembles exceptionnels en analyse de Fourier, Ann. Inst. Fourier (Grenoble) 18 (1968), no. 2, 31–71 (1969) (French). MR 412739, DOI 10.5802/aif.291
- J. R. Ringrose, Lectures on the trace in a finite von Neumann algebra, Lectures on operator algebras (Tulane Univ. Ring and Operator Theory Year, 1970–1971, Vol. II; dedicated to the memory of David M. Topping), Lecture Notes in Math., Vol. 247, Springer, Berlin, 1972, pp. 309–354. MR 0361816
- Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
- Wolfgang Ruppert, Compact semitopological semigroups: an intrinsic theory, Lecture Notes in Mathematics, vol. 1079, Springer-Verlag, Berlin, 1984. MR 762985, DOI 10.1007/BFb0073675
- Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
Additional Information
- Colin C. Graham
- Affiliation: Department of Mathematics, University of British Columbia, RR #1 – D-156, Bowen Island, British Columbia, Canada V0N 1G0
- Email: ccgraham@alum.mit.edu
- Anthony T. M. Lau
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
- MR Author ID: 110640
- Email: tlau@math.ualberta.ca
- Received by editor(s): January 23, 2003
- Received by editor(s) in revised form: December 8, 2004
- Published electronically: September 11, 2006
- Additional Notes: Both authors were partially supported by NSERC grants
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 1129-1160
- MSC (2000): Primary 43A15, 43A10; Secondary 46L10
- DOI: https://doi.org/10.1090/S0002-9947-06-04039-6
- MathSciNet review: 2262845