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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

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
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 $ U(z) = z$ 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 $ \mathcal{B}(H)$ (bounded operators on a Hilbert space H) or $ {l_\infty }(X)$ (bounded functions on X), then E is finite dimensional if and only if each $ \mathrm{weak}^*$-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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0567978-4
PII: S 0002-9939(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