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

Abstract

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 .

Anthropocene Geopolitics

“…Both X and X * fail the FPP. This because Pfitzner showed in [23] that every nonreflexive subspace of an M -embedded Banach space contains an asymptotic isometric copy of c 0 and every nonreflexive subspace of an Lembedded Banach space contains an asymptotic isometric copy ℓ 1 . From these properties flows the failure of the FPP.…”

The almost Daugavet property and translation-invariant subspaces

“…So Theorem 3.2 gives another proof of the fact that every non-reflexive subspace of L 1 [0, 1] or more generally every non-reflexive subspace of an L-embedded space fails the fixed point property (cf. [4,Theorem 1.4;16,Corollary 4]).…”

The Dual of a Non-reflexive L-embedded Banach Space Contains l∞Isometrically

“…Each normalized sequence (x n ) in an L-embedded Banach space that spans l 1 almost isomorphically contains a subsequence each of whose w *accumulation points in the bidual attains its norm on the dual unit ball. To see this let (x * n ) and (x pn ) be the sequences given by the theorem and by Simons' extraction lemma (see (13) above), let x s be a w * -accumulation point of the x pn and let x * = x * n ; then x * = 1 and on the one hand x s = 1 by [8] and on the other hand x s (x * ) = lim x * (x pn ) (13) = lim x * n (x pn )…”

“…This follows from [4, Cor. III.2.12] which states that there is an L-embedded Banach space which is the dual of an M-embedded space (to wit the dual of c 0 with an equivalent norm) which is strictly convex and therefore does not contain l 1 isometrically although it contains, as do all non-reflexive L-embedded spaces, l 1 asymptotically ( [8], see [3] for the definition of asymptotic copies and the difference to almost isometric ones).…”

“…IV.2.7]) that the dual of a non-reflexive L-embedded Banach space contains c 0 isomorphically. For a special class of L-embedded Banach spaces the construction of the c 0 -copy has been improved so to yield an asymptotic one ([8,Prop. 6]) and it turns out that this improvement is possible in the general case which together with Dowling's result yields isometric copies of c 0 in the dual of an L-embedded Banach space.…”