(1+)-complemented, (1+)-isomorphic copies of L1 in dual Banach spaces

• Published in: Archiv der Mathematik Vol. 119; no. 5; pp. 495 - 505
• Main Authors: Chen, Dongyang, Kania, Tomasz, Ruan, Yingbin
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2022; Springer Nature B.V
• Subjects: Banach spaces; Mathematics; Mathematics and Statistics; Quotient maps; Secondary 46C05; Primary 46B15; Banach spaces; Complemented subspaces; Isomorphic copies of
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-022-01778-2

The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on the Hagler–Stegall characterisation of dual spaces containing complemented copies of
L
1
. As a corollary, we obtain the following quantitative version of the Hagler–Stegall theorem asserting that for a Banach space
X
, the following statements are equivalent:
X
contains almost isometric contains almost isometric copies of
(
⨁
n
=
1
∞
ℓ
∞
n
)
ℓ
1
;
for all
ε
>
0
,
X
∗
contains a
(
1
+
ε
)
-complemented,
(
1
+
ε
)
-isomorphic copy of
L
1
;
for all
ε
>
0
,
X
∗
contains a
(
1
+
ε
)
-complemented,
(
1
+
ε
)
-isomorphic copy of
C
[
0
,
1
]
∗
.
Moreover, if
X
is separable, one may add the following assertion:
for all
ε
>
0
, there exists a
(
1
+
ε
)
-quotient map
T
:
X
→
C
(
Δ
)
so that
T
∗
[
C
(
Δ
)
∗
]
is
(
1
+
ε
)
-complemented in
X
∗
, where
Δ
is the Cantor set