Sets with fixed point property for isometric mappings
Author:
Anthony To Ming Lau
Journal:
Proc. Amer. Math. Soc. 79 (1980), 388392
MSC:
Primary 47H10
MathSciNet review:
567978
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Abstract: A subset K of a Banach space E is said to have the fixed point property for isometric mappings (f. p. p.) if there exists z in K such that for each isometric mapping U from K onto K. We prove that any bounded closed subsets of E with uniform relative normal structure have the f. p. p. We also prove that if E is either (bounded operators on a Hilbert space H) or (bounded functions on X), then E is finite dimensional if and only if each compact convex subset of E has the f. p. p. This is also equivalent to the convex set of (normal) states on E having the f. p. p.
 [1]
M.
S. Brodskiĭ and D.
P. Mil′man, On the center of a convex set, Doklady Akad.
Nauk SSSR (N.S.) 59 (1948), 837–840 (Russian). MR 0024073
(9,448f)
 [2]
Ralph
DeMarr, Common fixed points for commuting contraction
mappings, Pacific J. Math. 13 (1963),
1139–1141. MR 0159229
(28 #2446)
 [3]
Michael
Edelstein, On nonexpansive mappings of Banach spaces, Proc.
Cambridge Philos. Soc. 60 (1964), 439–447. MR 0164222
(29 #1521)
 [4]
, Remarks and questions concerning nonexpansive mappings, Fixed Point Theory and its Application, edited by S. Swaminathan, Academic Press, New York, 1976, pp. 6367.
 [5]
E.
Granirer, Extremely amenable semigroups, Math. Scand.
17 (1965), 177–197. MR 0197595
(33 #5760)
 [6]
E.
Granirer, Extremely amenable semigroups. II, Math. Scand.
20 (1967), 93–113. MR 0212551
(35 #3422)
 [7]
Frederick
P. Greenleaf, Invariant means on topological groups and their
applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand
Reinhold Co., New York, 1969. MR 0251549
(40 #4776)
 [8]
E. Hewitt and K. Ross, Abstract harmonic analysis. I, SpringerVerlag, Berlin, 1963.
 [9]
Theodore
Mitchell, Constant functions and left invariant
means on semigroups, Trans. Amer. Math.
Soc. 119 (1965),
244–261. MR 0193523
(33 #1743), http://dx.doi.org/10.1090/S00029947196501935238
 [10]
Shôichirô
Sakai, 𝐶*algebras and 𝑊*algebras,
SpringerVerlag, New York, 1971. Ergebnisse der Mathematik und ihrer
Grenzgebiete, Band 60. MR 0442701
(56 #1082)
 [11]
Paolo
M. Soardi, Existence of fixed points of
nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), no. 1, 25–29. MR 512051
(80c:47051), http://dx.doi.org/10.1090/S00029939197905120516
 [1]
 M. S. Brodskii and D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948), 838840. (Russian) MR 0024073 (9:448f)
 [2]
 R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 11391141. MR 0159229 (28:2446)
 [3]
 M. Edelstein, On nonexpansive mappings of Banach spaces, Proc. Cambridge Philos. Soc. 60 (1964), 439447. MR 0164222 (29:1521)
 [4]
 , Remarks and questions concerning nonexpansive mappings, Fixed Point Theory and its Application, edited by S. Swaminathan, Academic Press, New York, 1976, pp. 6367.
 [5]
 E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 117197. MR 0197595 (33:5760)
 [6]
 , Extremely amenable semigroups. II, Math. Scand. 20 (1967), 93113. MR 0212551 (35:3422)
 [7]
 F. P. Greenleaf, Invariant means on topological groups, Van Nostrand Math. Studies, no. 16, Van Nostrand Reinhold and Winston, Princeton, N. J., 1969. MR 0251549 (40:4776)
 [8]
 E. Hewitt and K. Ross, Abstract harmonic analysis. I, SpringerVerlag, Berlin, 1963.
 [9]
 T. Mitchell, Constant functions and leftinvariant means on semigroups, Trans. Amer. Math. Soc. 119 (1965), 244261. MR 0193523 (33:1743)
 [10]
 S. Sakai, algebras and algebras, SpringerVerlag, Berlin, 1971. MR 0442701 (56:1082)
 [11]
 P. M. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach spaces, Proc. Amer. Math. Soc. 73 (1979), 2529. MR 512051 (80c:47051)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198005679784
PII:
S 00029939(1980)05679784
Keywords:
Uniform relative normal structure,
isometric mappings,
fixed point property,
invariant means on groups,
operators on Hilbert space
Article copyright:
© Copyright 1980 American Mathematical Society
