Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces
Commentationes Mathematicae, Tome 56 (2016) no. 1.
Voir la notice de l'article dans European Digital Mathematics Library
Let
X
be a Banach space and
𝒮
𝑒𝑞
(
X
*
*
)
(resp.,
X
ℵ
0
) the subset of elements
ψ
∈
X
*
*
such that there exists a sequence
(
x
n
)
n
≥
1
⊂
X
such that
x
n
→
ψ
in the
w
*
-topology of
X
*
*
(resp., there exists a separable subspace
Y
⊂
X
such that
ψ
∈
Y
¯
w
*
). Then: (i) if
Dens
(
X
)
≥
ℵ
1
, the property
X
*
*
=
X
ℵ
0
(resp.,
X
*
*
=
𝒮
𝑒𝑞
(
X
*
*
)
) is
ℵ
1
-determined, i.e.,
X
has this property iff
Y
has, for every subspace
Y
⊂
X
with
Dens
(
Y
)
=
ℵ
1
; (ii) if
X
*
*
=
X
ℵ
0
,
(
B
(
X
*
*
)
,
w
*
)
has countable tightness; (iii) under the Martin’s axiom
𝑀𝐴
(
ω
1
)
we have
X
*
*
=
𝒮
𝑒𝑞
(
X
*
*
)
iff
(
B
(
X
*
)
,
w
*
)
has countable tightness and
o
v
e
r
l
i
n
e
co
(
B
)
=
co
¯
w
*
(
K
)
for every subspace
Y
⊂
X
, every
w
*
-compact subset
K
of
Y
*
, and every boundary
B
⊂
K
.
Mots-clés :
Boundaries, Martin’s Axiom, equality $Seq(X^{**})=X^{**}$, super-(P) property
@article{COMA_2016__56_1_292437,
author = {Antonio S. Granero and Juan M. Hern\'andez},
title = {Boundaries, {Martin's} {Axiom,} and {(P)-properties} in dual {Banach} spaces},
journal = {Commentationes Mathematicae},
publisher = {mathdoc},
volume = {56},
number = {1},
year = {2016},
language = {en},
url = {https://geodesic-test.mathdoc.fr/item/COMA_2016__56_1_292437/}
}
TY - JOUR AU - Antonio S. Granero AU - Juan M. Hernández TI - Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces JO - Commentationes Mathematicae PY - 2016 VL - 56 IS - 1 PB - mathdoc UR - https://geodesic-test.mathdoc.fr/item/COMA_2016__56_1_292437/ LA - en ID - COMA_2016__56_1_292437 ER -
Antonio S. Granero; Juan M. Hernández. Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces. Commentationes Mathematicae, Tome 56 (2016) no. 1. https://geodesic-test.mathdoc.fr/item/COMA_2016__56_1_292437/