Reflexivity and the Fixed-Point Property for Nonexpansive Maps

Dowling, Patrick N.; Lennard, Chris; Turett, B.

doi:10.1006/jmaa.1996.0229

Cited by 33 publications

(17 citation statements)

References 6 publications

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“…Following [4,5,6] we say that (x n ) spans l 1 asymptotically isometrically (or just that (x n ) spans l 1 asymptotically) if there is a sequence (δ n ) in [0, 1[ tending to 0 such that…”

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confidence: 99%

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A note on asymptotically isometric copies of $l^1$ and $c_0$

Pfitzner¹

2000

Proc. Amer. Math. Soc.

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Abstract. Nonreflexive Banach spaces that are complemented in their bidual by an L-projection-like preduals of von Neumann algebras or the Hardy space H 1 -contain, roughly speaking, many copies of l 1 which are very close to isometric copies. Such l 1 -copies are known to fail the fixed point property. Similar dual results hold for c 0 .In [4] it is shown that an isomorphic l 1 -copy does not necessarily contain asymptotically isometric l 1 -copies although by James' classical distortion theorem it always contains almost isomorphic In the present note we modify a construction of Godefroy in order to show that every nonreflexive subspace of any L-embedded Banach space contains an asymptotic l 1 -copy and thus, in particular, fails the fixed point property. Analogous results hold for c 0 and M-embedded spaces.Let (x n ) be a sequence of nonzero elements in a Banach space X. We say that (x n ) spans l 1 r-isomorphically or just isomorphically if there existsTrivially the property of spanning l 1 almost isometrically passes to subsequences. Analogously, we say that (x n ) spans c 0 almost isometrically if there exists a sequence (δ m ) as above such that (1 − δ m ) sup m≤n≤m |α n | ≤ m n=m α n x n ≤ (1 + δ m ) sup m≤n≤m |α n | for all m ≤ m .

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“…Following [4,5,6] we say that (x n ) spans l 1 asymptotically isometrically (or just that (x n ) spans l 1 asymptotically) if there is a sequence (δ n ) in [0, 1[ tending to 0 such that…”

mentioning

confidence: 99%

“…It is now immediate that every nonreflexive subspace of an M-embedded space fails the fixed point property (see [6, Prop. 7]).…”

mentioning

confidence: 99%

A note on asymptotically isometric copies of $l^1$ and $c_0$

Pfitzner¹

2000

Proc. Amer. Math. Soc.

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“…Theorem 1.2 (see [3] or [11]). If a Banach space X contains an asymptotically isometric copy of c 0 , then X fails the fixed point property for nonexpansive (and even contractive) mappings on bounded, closed and convex subsets of X. Theorem 1.3 (see [5] or [11]).…”

Section: Theorem 11 (James's Distortion Theorem Stronger Version) mentioning

confidence: 99%

Renormings of c0 and the minimal displacement problem

Piasecki¹

2014

Annales UMCS, Mathematica

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“…The notion of asymptotically isometric copy of c 0 was introduced in [6], where it is shown that if a Banach space X contains such a copy, then X fails the fixed-point property for nonexpansive self-mappings on closed bounded convex subsets of X.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically isometric copies of c_0 in Musielak-Orlicz spaces

Narloch¹,

Szymaszkiewicz²

2014

OpMath

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Abstract. Criteria in order that a Musielak-Orlicz function space L Φ as well as Musielak-Orlicz sequence space l Φ contains an asymptotically isometric copy of c0 are given. These results extend some results of [Y.A. Cui, H. Hudzik, G. Lewicki, Order asymptotically isometric copies of c0 in the subspaces of order continuous elements in Orlicz spaces, Journal of Convex Analysis 21 (2014)] to Musielak-Orlicz spaces.

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Reflexivity and the Fixed-Point Property for Nonexpansive Maps

Cited by 33 publications

References 6 publications

A note on asymptotically isometric copies of $l^1$ and $c_0$

A note on asymptotically isometric copies of $l^1$ and $c_0$

Renormings of c0 and the minimal displacement problem

Asymptotically isometric copies of c_0 in Musielak-Orlicz spaces

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