Sets with fixed point property for isometric mappings

Author:
Anthony To Ming Lau

Journal:
Proc. Amer. Math. Soc. **79** (1980), 388-392

MSC:
Primary 47H10

DOI:
https://doi.org/10.1090/S0002-9939-1980-0567978-4

MathSciNet review:
567978

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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. Brodskii and D. P. Milman,*On the center of a convex set*, Dokl. Akad. Nauk SSSR**59**(1948), 838-840. (Russian) MR**0024073 (9:448f)****[2]**R. DeMarr,*Common fixed points for commuting contraction mappings*, Pacific J. Math.**13**(1963), 1139-1141. MR**0159229 (28:2446)****[3]**M. Edelstein,*On non-expansive mappings of Banach spaces*, Proc. Cambridge Philos. Soc.**60**(1964), 439-447. MR**0164222 (29:1521)****[4]**-,*Remarks and questions concerning non-expansive mappings*, Fixed Point Theory and its Application, edited by S. Swaminathan, Academic Press, New York, 1976, pp. 63-67.**[5]**E. Granirer,*Extremely amenable semigroups*, Math. Scand.**17**(1965), 117-197. MR**0197595 (33:5760)****[6]**-,*Extremely amenable semigroups*. II, Math. Scand.**20**(1967), 93-113. 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, Springer-Verlag, Berlin, 1963.**[9]**T. Mitchell,*Constant functions and left-invariant means on semigroups*, Trans. Amer. Math. Soc.**119**(1965), 244-261. MR**0193523 (33:1743)****[10]**S. Sakai, -*algebras and*-*algebras*, Springer-Verlag, Berlin, 1971. MR**0442701 (56:1082)****[11]**P. M. Soardi,*Existence of fixed points of non-expansive mappings in certain Banach spaces*, Proc. Amer. Math. Soc.**73**(1979), 25-29. MR**512051 (80c:47051)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47H10

Retrieve articles in all journals with MSC: 47H10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0567978-4

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