Fixed point property and the Fourier algebra of a locally compact group
HTML articles powered by AMS MathViewer
- by Anthony To-Ming Lau and Michael Leinert PDF
- Trans. Amer. Math. Soc. 360 (2008), 6389-6402 Request permission
Abstract:
We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) $\mathcal {L}^1$ spaces and use this for the Fourier algebra $A(G)$ of a locally compact group $G.$ In particular we show that if $G$ is an IN-group, then $A(G)$ has the weak fpp if and only if $G$ is compact. We also show that if $G$ is any locally compact group, then $A(G)$ has the fixed point property (fpp) if and only if $G$ is finite. Furthermore if a nonzero closed ideal of $A(G)$ has the fpp, then $G$ must be discrete.References
- Dale E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), no. 3, 423–424. MR 612733, DOI 10.1090/S0002-9939-1981-0612733-0
- Larry Baggett and Keith Taylor, Groups with completely reducible regular representation, Proc. Amer. Math. Soc. 72 (1978), no. 3, 593–600. MR 509261, DOI 10.1090/S0002-9939-1978-0509261-X
- T. Domínguez Benavides, M. A. Japón Pineda, and S. Prus, Weak compactness and fixed point property for affine mappings, J. Funct. Anal. 209 (2004), no. 1, 1–15. MR 2039215, DOI 10.1016/j.jfa.2002.02.001
- T. Domínguez Benavides and María A. Japón Pineda, Fixed points of nonexpansive mappings in spaces of continuous functions, Proc. Amer. Math. Soc. 133 (2005), no. 10, 3037–3046. MR 2159783, DOI 10.1090/S0002-9939-05-08149-9
- Richard D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. MR 704815, DOI 10.1007/BFb0069321
- Marek Bożejko, The existence of $\Lambda (p)$ sets in discrete noncommutative groups, Boll. Un. Mat. Ital. (4) 8 (1973), 579–582 (English, with Italian summary). MR 0344805
- Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
- Ronald E. Bruck Jr., A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J. Math. 53 (1974), 59–71. MR 361945
- Cho-Ho Chu, A note on scattered $C^{\ast }$-algebras and the Radon-Nikodým property, J. London Math. Soc. (2) 24 (1981), no. 3, 533–536. MR 635884, DOI 10.1112/jlms/s2-24.3.533
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
- Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
- P. N. Dowling, C. J. Lennard, and B. Turett, The fixed point property for subsets of some classical Banach spaces, Nonlinear Anal. 49 (2002), no. 1, Ser. A: Theory Methods, 141–145. MR 1887917, DOI 10.1016/S0362-546X(01)00104-3
- D. van Dulst and Brailey Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), Banach space theory and its applications (Bucharest, 1981) Lecture Notes in Math., vol. 991, Springer, Berlin-New York, 1983, pp. 35–43. MR 714171
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
- R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), no. 4, 743–749. MR 595102, DOI 10.1216/RMJ-1980-10-4-743
- W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
- Kazimierz Goebel and W. A. Kirk, Classical theory of nonexpansive mappings, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 49–91. MR 1904274
- Hideki Kosaki, Applications of the complex interpolation method to a von Neumann algebra: noncommutative $L^{p}$-spaces, J. Funct. Anal. 56 (1984), no. 1, 29–78. MR 735704, DOI 10.1016/0022-1236(84)90025-9
- Anthony To Ming Lau and Peter F. Mah, Normal structure in dual Banach spaces associated with a locally compact group, Trans. Amer. Math. Soc. 310 (1988), no. 1, 341–353. MR 937247, DOI 10.1090/S0002-9947-1988-0937247-0
- Anthony To Ming Lau and Peter F. Mah, Quasinormal structures for certain spaces of operators on a Hilbert space, Pacific J. Math. 121 (1986), no. 1, 109–118. MR 815037
- Anthony To-Ming Lau, Peter F. Mah, and Ali Ülger, Fixed point property and normal structure for Banach spaces associated to locally compact groups, Proc. Amer. Math. Soc. 125 (1997), no. 7, 2021–2027. MR 1372037, DOI 10.1090/S0002-9939-97-03773-8
- Anthony To Ming Lau and Ali Ülger, Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems, Trans. Amer. Math. Soc. 337 (1993), no. 1, 321–359. MR 1147402, DOI 10.1090/S0002-9947-1993-1147402-7
- Michael Leinert, On integration with respect to a trace, Aspects of positivity in functional analysis (Tübingen, 1985) North-Holland Math. Stud., vol. 122, North-Holland, Amsterdam, 1986, pp. 231–239. MR 859732
- Michael Leinert, Integration und Maß, Friedr. Vieweg & Sohn, Braunschweig, 1995 (German, with German summary). MR 1396785
- Chris Lennard, ${\scr C}_1$ is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), no. 1, 71–77. MR 943795, DOI 10.1090/S0002-9939-1990-0943795-4
- Teck Cheong Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980), no. 1, 135–143. MR 599326
- B. Maurey, Points fixes des contractions de certains faiblement compacts de $L^{1}$, Seminar on Functional Analysis, 1980–1981, École Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19 (French). MR 659309
- Paul S. Mostert, Sections in principal fibre spaces, Duke Math. J. 23 (1956), 57–71. MR 75575
- Edward Nelson, Notes on non-commutative integration, J. Functional Analysis 15 (1974), 103–116. MR 0355628, DOI 10.1016/0022-1236(74)90014-7
- T. W. Palmer, Classes of nonabelian, noncompact, locally compact groups, Rocky Mountain J. Math. 8 (1978), no. 4, 683–741. MR 513952, DOI 10.1216/RMJ-1978-8-4-683
- Massimo Angelo Picardello, Lacunary sets in discrete noncommutative groups, Boll. Un. Mat. Ital. (4) 8 (1973), 494–508 (English, with Italian summary). MR 0344804
- I. E. Segal, Equivalences of measure spaces, Amer. J. Math. 73 (1951), 275–313. MR 41191, DOI 10.2307/2372178
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
- Keith F. Taylor, Geometry of the Fourier algebras and locally compact groups with atomic unitary representations, Math. Ann. 262 (1983), no. 2, 183–190. MR 690194, DOI 10.1007/BF01455310
- Kôsaku Yosida, Functional analysis, 5th ed., Grundlehren der Mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin-New York, 1978. MR 0500055
Additional Information
- Anthony To-Ming 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
- Michael Leinert
- Affiliation: Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld, Gebäude 294, 69120 Heidelberg, Germany
- Email: leinert@math.uni-heidelberg.de
- Received by editor(s): November 10, 2006
- Published electronically: July 22, 2008
- Additional Notes: The research of the first author was supported by NSERC Grant A-7679
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 6389-6402
- MSC (2000): Primary 43A15, 47A09; Secondary 43A20, 47H10, 46B22
- DOI: https://doi.org/10.1090/S0002-9947-08-04622-9
- MathSciNet review: 2434292