The fixed point property via dual space properties

Abstract

A Banach space has the weak fixed point property if its dual space has a weak * sequentially compact unit ball and the dual space satisfies the weak * uniform Kadec-Klee property; and it has the fixed point property if there exists ε > 0 such that, for every infinite subset A of the unit sphere of the dual space, A ∪ (−A) fails to be (2 − ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.Determining conditions on a Banach space X so that every nonexpansive mapping from a nonempty, closed, bounded, convex subset of X into itself has a fixed point has been of considerable interest for many years. A Banach space has the fixed point property if, for each nonempty, closed, bounded, convex subset C of X, every nonexpansive mapping of C into itself has a fixed point. A Banach space is said to have the weak fixed point property if the class of sets C above is restricted to the set of weakly compact convex sets; and a Banach space is said to have the weak * fixed point property if X is a dual space and the class of sets C is restricted to the set of weak * compact convex subsets of X.A well-known open problem in Banach spaces is whether every reflexive Banach space has the fixed point property for nonexpansive mappings. The question of whether more restrictive

Fixed point property for Banach algebras associated to locally compact groups

“…See also the note on page 775 in [10]. Now if H is any Hilbert space, and Y is any separable closed subspace of K(H), then as the proof of Theorem 5 in [9] shows, every separable subspace Y of K(H) can be embedded as a subspace of K(H 0 ), where H 0 is a separable Hilbert subspace of H. Thus Y and hence K(H) has the weak fixed point property.…”

Fixed point properties of C∗-algebras

“…Proof. (1) was proved by Maurey [25], (2) follows from results of Lennard [23] combined with the results of Dowling, Randrianantoanina and Turett [12], (3) was proved by Elton, Lin, Odell and Szarek [15]. 2…”

“…(2008) Dowling, Randrianantoanina and Turett [12] proved that if (i) E * is weak * sequentially compact, and (ii) E * has the weak * uniform Kadec-Klee property, then E has the w-fpp, answering a question posed by Lau, Mah and Ülger in [22].…”

A product space with the fixed point property

“…[17,16]. If X* is O-convex, that is if X is E-convex, then the Banach space X has the fixed point property for nonexpansive mappings.…”