Abstract
Suppose that Q is a hbox {weak}^{*} compact convex subset of a dual Banach space with the Radon–Nikodým property. We show that if (S, Q) is a nonexpansive and norm-distal dynamical system, then there is a fixed point of S in Q and the set of fixed points is a nonexpansive retract of Q. As a consequence we obtain a nonlinear extension of the Bader–Gelander–Monod theorem concerning isometries in L-embedded Banach spaces. A similar statement is proved for weakly compact convex subsets of a locally convex space, thus giving the nonlinear counterpart of the Ryll-Nardzewski theorem.
Highlights
Fixed point theorems for groups and semigroups of mappings provide a powerful tool in diverse branches of mathematics
The recent Bader–Gelander–Monod theorem for affine isometries preserving a bounded set in L-embedded Banach spaces has several applications in group theory and in the theory of operator algebras, including the optimal solution to the old “derivation problem” in L1(G)
One of the first general fixed point theorems for isometries, next to Kakutani-type theorems, was shown by Brodskiı and Mil’man in [6] –all surjective isometries acting on a weakly compact, convex subset of a Banach space with normal structure have a common fixed point
Summary
Fixed point theorems for groups and semigroups of mappings provide a powerful tool in diverse branches of mathematics. Kakutani-type theorems have found numerous applications in functional analysis, harmonic analysis and ergodic theory. Furstenberg’s structure theorem and its consequences are fundamental tools in topological dynamics. The Bruhat–Tits theorem concerning complete metric spaces satisfying the parallelogram law turns out to be useful in differential geometry. Kazhdan’s property (T) plays a prominent role in geometric group theory and related fields. The recent Bader–Gelander–Monod theorem for affine isometries preserving a bounded set in L-embedded Banach spaces has several applications in group theory and in the theory of operator algebras, including the optimal solution to the old “derivation problem” in L1(G). Dedicated to Professor Kazimierz Goebel on the occasion of his 80th birthday
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