Every nonreflexive subspace of $L_1[0,1]$ fails the fixed point property

Dowling, Patrick N.; Lennard, Chris

doi:10.1090/s0002-9939-97-03577-6

Cited by 57 publications

(24 citation statements)

References 10 publications

(2 reference statements)

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“…Maurey's method was applied by P.-K. Lin [37] who proved that every Banach space with a 1-unconditional basis enjoys WFPP. In 1997, P. Dowling and C. Lennard [17] proved that every nonreflexive subspace of L 1 [0, 1] fails FPP and they developed their techniques in a series of papers. For a fuller discussion of metric fixed point theory we refer the reader to [3,26,28,31].…”

mentioning

confidence: 99%

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

Prus

Wiśnicki

2008

Studia Math.

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confidence: 99%

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

Prus

Wiśnicki

2008

Studia Math.

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“…A Banach space, which is very similar to ℓ 1 and offers exactly the same features, but without a norm obtained only by a linear expansion of the usual one, is rarely included in the literature (even the renorming of Here, we can note that Maurey [23] showed that (c 0 , ∥ • ∥ ∞ ) and reflexive subspaces of L 1 [0, 1] do have w-fpp using ultrafilter techniques. Conversely to Maurey's result, Downling and Lennard [8] showed that every nonreflexive subspace of L 1 [0, 1] fails the fixed point property. Before their result, in 1996, by Carothers et al…”

Section: Introductionmentioning

confidence: 85%

Fixed point properties for a degenerate Lorentz–Marcinkiewicz space

Nezir¹

2019

Turk J Math

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We construct an equivalent renorming of ℓ 1 , which turns out to produce a degenerate ℓ 1-analog Lorentz-Marcinkiewicz space ℓ δ,1 , where the weight sequence δ = (δn) n∈N = (2, 1, 1, 1, • • •) is a decreasing positive sequence in ℓ ∞ \c0 , rather than in c0\ℓ 1 (the usual Lorentz situation). Then we obtain its isometrically isomorphic predual ℓ 0 δ,∞ and dual ℓ δ,∞ , corresponding degenerate c0-analog and ℓ ∞-analog Lorentz-Marcinkiewicz spaces, respectively. We prove that both spaces ℓ δ,1 and ℓ 0 δ,∞ enjoy the weak fixed point property (w-fpp) for nonexpansive mappings yet they fail to have the fixed point property (fpp) for nonexpansive mappings since they contain an asymptotically isometric copy of ℓ 1 and c0 , respectively. In fact, we prove for both spaces that there exist nonempty, closed, bounded, and convex subsets with invariant fixed point-free affine, nonexpansive mappings on them and so they fail to have fpp for affine nonexpansive mappings. Also, we show that any nonreflexive subspace of l 0 δ,∞ contains an isomorphic copy of c0 and so fails fpp for strongly asymptotically nonexpansive maps. Finally, we get a Goebel and Kuczumow analogy by proving that there exists an infinite dimensional subspace of ℓ δ,1 with fpp for affine nonexpansive mappings.

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“…Using the main decomposition for the case p = 1, we can conclude that every nonreflexive subspace of duals of C * -algebras contains sequences that generate complemented copies of 1 and are asymptotically isometric. As in [9] and [10], these asymptotically isometric copies of 1 yield self maps on convex bounded sets that fail to have any fixed points. These lead to a more general structural consequence that non-reflexive subspaces of duals of JB * -triples fail the fixed point property for self-maps on closed bounded convex sets.…”

Section: Introductionmentioning

confidence: 90%

“…A result of Maurey ([23], see also [11]) states that every reflexive subspace of L 1 [0, 1] has the fixed point property for nonexpansive mappings (FPP). Later, Dowling and Lennard showed that the converse of Maurey's result is valid: every nonreflexive subspace of L 1 [0, 1] fails the FPP ( [10]). This section is for the study of generalizations to the case of duals of C * -algebras and requires the notion of asymptotically isometric copies of 1 which was introduced by Dowling and Lennard in [10].…”

Section: Applicationsmentioning

confidence: 99%

Kadec–Pełczyński decomposition for Haagerup L^p-spaces

Randrianantoanina

2002

Math. Proc. Camb. Phil. Soc.

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Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in Lp[0, 1] to the case of the Haagerup Lp-spaces (1 [les ] p < 1 ). In particular, we prove that if { φn}∞n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φnk}∞k=1 of {φn}∞n=1, a decomposition φnk = yk+zk such that {yk, k [ges ] 1} is relatively weakly compact and the support projections supp(zk) ↓k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]1 and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.

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Every nonreflexive subspace of $L_1[0,1]$ fails the fixed point property

Cited by 57 publications

References 10 publications

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

Fixed point properties for a degenerate Lorentz–Marcinkiewicz space

Kadec–Pełczyński decomposition for Haagerup L^p-spaces

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Every nonreflexive subspace of $L_1[0,1]$ fails the fixed point property

Cited by 57 publications

References 10 publications

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

On the fixed point property in direct sums of Banach spaces with strictly monotone norms

Fixed point properties for a degenerate Lorentz–Marcinkiewicz space

Kadec–Pełczyński decomposition for Haagerup Lp-spaces

Contact Info

Product

Resources

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Kadec–Pełczyński decomposition for Haagerup L^p-spaces