TY - JOUR
T1 - Separable lifting property and extensions of local reflexivity
AU - Johnson, William B.
AU - Oikhberg, Timur
PY - 2001
Y1 - 2001
N2 - A Banach space X is said to have the separable lifting property if for every subspace Y of X** containing X and such that Y/X is separable there exists a bounded linear lifting from Y/X to Y. We show that if a sequence of Banach spaces E
1, E
2,... has the joint uniform approximation property and E
n is c-complemented in E
n** for every n (with c fixed), then (∑
n E
n)
0 has the separable lifting property. In particular, if E
n is a ℒ
pn,λ-space for every n (1 < p
n < ∞, λ independent of n), an L
∞ or an L
1 space, then (∑
n E
n)
0 has the separable lifting property. We also show that there exists a Banach space X which is not extendably locally reflexive; moreover, for every n there exists an n-dimensional subspace E → X** such that if u: X** → X** is an operator (= bounded linear operator) such that u(E) ⊂ X, then ∥(u|
E)
-1∥ · ∥u∥ ≥ c√n, where c is a numerical constant.
AB - A Banach space X is said to have the separable lifting property if for every subspace Y of X** containing X and such that Y/X is separable there exists a bounded linear lifting from Y/X to Y. We show that if a sequence of Banach spaces E
1, E
2,... has the joint uniform approximation property and E
n is c-complemented in E
n** for every n (with c fixed), then (∑
n E
n)
0 has the separable lifting property. In particular, if E
n is a ℒ
pn,λ-space for every n (1 < p
n < ∞, λ independent of n), an L
∞ or an L
1 space, then (∑
n E
n)
0 has the separable lifting property. We also show that there exists a Banach space X which is not extendably locally reflexive; moreover, for every n there exists an n-dimensional subspace E → X** such that if u: X** → X** is an operator (= bounded linear operator) such that u(E) ⊂ X, then ∥(u|
E)
-1∥ · ∥u∥ ≥ c√n, where c is a numerical constant.
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U2 - 10.1215/ijm/1258138258
DO - 10.1215/ijm/1258138258
M3 - Article
SN - 0019-2082
VL - 45
SP - 123
EP - 137
JO - Illinois Journal of Mathematics
JF - Illinois Journal of Mathematics
IS - 1
ER -