Fixed point property and the Fourier algebra of a locally compact group. (English) Zbl 1153.43008
A non-empty, closed, bounded, convex subset \(K\) of a Banach space \(E\) is said to have the fixed point property (short: fpp) if every non-expansive map of \(K\) into itself has a fixed point. We say that \(E\) has the fpp if every closed, bounded, convex subset of \(E\) has the fpp, and that \(E\) has the weak fpp if every weakly compact convex subset of \(E\) has the fpp. For instance, \(\ell^1\) has the weak fpp, but lacks the fpp, whereas \(L^1[0,1]\) does not have the weak fpp (and thus not the fpp either).
In the paper under review, the authors first give characterizations of the weak fpp for certain commutative and non-commutative \(L^1\)-spaces.
They then focus on the Fourier algebra \(A(G)\) of a locally compact group. Among their results are the following: \(A(G)\) has the fpp if and only if \(G\) is finite, and, provided \(G\) is an \([\text{IN}]\)-group, \(A(G)\) has the weak fpp if and only if \(G\) is compact.
In the paper under review, the authors first give characterizations of the weak fpp for certain commutative and non-commutative \(L^1\)-spaces.
They then focus on the Fourier algebra \(A(G)\) of a locally compact group. Among their results are the following: \(A(G)\) has the fpp if and only if \(G\) is finite, and, provided \(G\) is an \([\text{IN}]\)-group, \(A(G)\) has the weak fpp if and only if \(G\) is compact.
Reviewer: Volker Runde (Edmonton)
MSC:
| 43A99 | Abstract harmonic analysis |
| 43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |
| 47H10 | Fixed-point theorems |
| 43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
| 46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |
Keywords:
fixed point property; weak fixed point property; (non-commutative) \(L^1\)-space; Fourier algebraReferences:
| [1] | Dale E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), no. 3, 423 – 424. · Zbl 0468.47036 |
| [2] | Larry Baggett and Keith Taylor, Groups with completely reducible regular representation, Proc. Amer. Math. Soc. 72 (1978), no. 3, 593 – 600. · Zbl 0403.22007 |
| [3] | 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. · Zbl 1062.46017 · doi:10.1016/j.jfa.2002.02.001 |
| [4] | 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. · Zbl 1081.47054 |
| [5] | Richard D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodým property, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. · Zbl 0512.46017 |
| [6] | Marek Bożejko, The existence of \Lambda (\?) sets in discrete noncommutative groups, Boll. Un. Mat. Ital. (4) 8 (1973), 579 – 582 (English, with Italian summary). · Zbl 0279.43010 |
| [7] | Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041 – 1044. · Zbl 0128.35801 |
| [8] | Ronald E. Bruck Jr., A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J. Math. 53 (1974), 59 – 71. · Zbl 0312.47045 |
| [9] | Cho-Ho Chu, A note on scattered \?*-algebras and the Radon-Nikodým property, J. London Math. Soc. (2) 24 (1981), no. 3, 533 – 536. · Zbl 0438.46041 · doi:10.1112/jlms/s2-24.3.533 |
| [10] | J. Diestel and J. J. Uhl Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. · Zbl 0369.46039 |
| [11] | Jacques Dixmier, \?*-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. · Zbl 0372.46058 |
| [12] | 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. · Zbl 1008.47053 · doi:10.1016/S0362-546X(01)00104-3 |
| [13] | 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. · Zbl 0512.46015 |
| [14] | Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181 – 236 (French). · Zbl 0169.46403 |
| [15] | Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. · Zbl 0040.16802 |
| [16] | 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. · Zbl 0416.43001 |
| [17] | R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), no. 4, 743 – 749. · Zbl 0505.46011 · doi:10.1216/RMJ-1980-10-4-743 |
| [18] | W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004 – 1006. · Zbl 0141.32402 · doi:10.2307/2313345 |
| [19] | 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. · Zbl 1035.47033 |
| [20] | Hideki Kosaki, Applications of the complex interpolation method to a von Neumann algebra: noncommutative \?^{\?}-spaces, J. Funct. Anal. 56 (1984), no. 1, 29 – 78. · Zbl 0604.46063 · doi:10.1016/0022-1236(84)90025-9 |
| [21] | 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. · Zbl 0706.43003 |
| [22] | 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. · Zbl 0538.46011 |
| [23] | 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. · Zbl 0868.43001 |
| [24] | 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. · Zbl 0778.43003 |
| [25] | 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. · Zbl 0613.46054 |
| [26] | Michael Leinert, Integration und Maß, Friedr. Vieweg & Sohn, Braunschweig, 1995 (German, with German summary). · Zbl 0846.28001 |
| [27] | Chris Lennard, \?\(_{1}\) is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), no. 1, 71 – 77. · Zbl 0758.46017 |
| [28] | Teck Cheong Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980), no. 1, 135 – 143. · Zbl 0454.47046 |
| [29] | B. Maurey, Points fixes des contractions de certains faiblement compacts de \?\textonesuperior , Seminar on Functional Analysis, 1980 – 1981, École Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19 (French). |
| [30] | Paul S. Mostert, Sections in principal fibre spaces, Duke Math. J. 23 (1956), 57 – 71. · Zbl 0072.18102 |
| [31] | Edward Nelson, Notes on non-commutative integration, J. Functional Analysis 15 (1974), 103 – 116. · Zbl 0292.46030 |
| [32] | T. W. Palmer, Classes of nonabelian, noncompact, locally compact groups, Rocky Mountain J. Math. 8 (1978), no. 4, 683 – 741. · Zbl 0396.22001 · doi:10.1216/RMJ-1978-8-4-683 |
| [33] | Massimo Angelo Picardello, Lacunary sets in discrete noncommutative groups, Boll. Un. Mat. Ital. (4) 8 (1973), 494 – 508 (English, with Italian summary). · Zbl 0279.43009 |
| [34] | I. E. Segal, Equivalences of measure spaces, Amer. J. Math. 73 (1951), 275 – 313. · Zbl 0042.35502 · doi:10.2307/2372178 |
| [35] | Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. · Zbl 0436.46043 |
| [36] | Keith F. Taylor, Geometry of the Fourier algebras and locally compact groups with atomic unitary representations, Math. Ann. 262 (1983), no. 2, 183 – 190. · Zbl 0488.43009 · doi:10.1007/BF01455310 |
| [37] | Kôsaku Yosida, Functional analysis, 5th ed., Springer-Verlag, Berlin-New York, 1978. Grundlehren der Mathematischen Wissenschaften, Band 123. |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
