The weak stability of the positive face in 𝐿¹
Authors: Hu Zhibao 
Published: June 2010

Let F be the positive face of the unit ball of

L
1

[
0
,
1
]

{L^1}[0,1]

. We show that F is weakly stable in the sense that the midpoint map

Φ

1

/

2

:
F
×
F
→
F

{\Phi _{1/2}}:F \times F \to F

, with

Φ

1

/

2

(
f
,
g
)
=

1
2

(
f
+
g
)

{\Phi _{1/2}}(f,g) = \frac {1}{2}(f + g)

, is open with respect to the weak topology. This weak stability of the set F is the reason behind the fact that the notions of "huskable" and "strongly regular" operators coincide for operators from

L
1

[
0
,
1
]

{L^1}[0,1]

to a Banach space X. We prove this stability by showing that if

f
1

,

f
2

∈
F
,
λ
∈
(
0
,
1
)
,
ε
>
0

{f_1},{f_2} \in F,\lambda \in (0,1),\varepsilon > 0

and

δ
≥
max
{
2
ε

/

λ
,
2
ε

/

(
1
−
λ
)
}

\delta \geq \max \{ 2\varepsilon /\lambda ,2\varepsilon /(1 - \lambda )\}

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda upper V Subscript upper P comma delta Baseline left-parenthesis f 1 right-parenthesis plus left-parenthesis 1 minus lamda right-parenthesis upper V Subscript upper P comma delta Baseline left-parenthesis f 2 right-parenthesis superset-of upper V Subscript upper P comma epsilon Baseline left-bracket lamda f 1 plus left-parenthesis 1 minus lamda right-parenthesis f 2 right-bracket comma">

λ

V

P
,
δ

(

f
1

)
+
(
1
−
λ
)

V

P
,
δ

(

f
2

)
⊃

V

P
,
ε

[
λ

f
1

+
(
1
−
λ
)

f
2

]
,

\lambda {V_{P,\delta }}({f_1}) + (1 - \lambda ){V_{P,\delta }}({f_2}) \supset {V_{P,\varepsilon }}[\lambda {f_1} + (1 - \lambda ){f_2}],

\]
where

P
=
{

A
1

,
…
,

A
n

}

P = \{ {A_1}, \ldots ,{A_n}\}

is a finite positive partition of [0, 1] and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Subscript upper P comma epsilon Baseline left-parenthesis f right-parenthesis equals StartSet g element-of upper F colon sigma-summation Underscript i equals 1 Overscript n Endscripts StartAbsoluteValue integral Underscript upper A Subscript i Baseline Endscripts left-parenthesis f minus g right-parenthesis left-parenthesis t right-parenthesis d mu left-parenthesis t right-parenthesis EndAbsoluteValue less-than-or-equal-to epsilon EndSet">

V

P
,
ε

(
f
)
=

{

g
∈
F
:

∑

i
=
1

n

|

∫

A
i

(
f
−
g
)
(
t
)
d
μ
(
t
)

|

≤
ε

}

{V_{P,\varepsilon }}(f) = \left \{ {g \in F:\sum \limits _{i = 1}^n {\left | {\int _{{A_i}} {(f - g)(t)d\mu (t)} } \right | \leq \varepsilon } } \right \}

\]
for any f in F. We construct an example showing that for any

0
>
λ
>
1

0 > \lambda > 1

there are functions

f
1

{f_1}

and

f
2

{f_2}

in F such that if

0
>
ε
>
2
min
{
λ
,
1
−
λ
}

0 > \varepsilon > 2\min \{ \lambda ,1 - \lambda \}

and

0
≤
δ
>
max
{
ε

/

λ
,
ε

/

(
1
−
λ
)
}

0 \leq \delta > \max \{ \varepsilon /\lambda ,\varepsilon /(1 - \lambda )\}

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda upper V Subscript upper P comma delta Baseline left-parenthesis f 1 right-parenthesis plus left-parenthesis 1 minus lamda right-parenthesis upper V Subscript upper P comma delta Baseline left-parenthesis f 2 right-parenthesis not-a-superset-of upper V Subscript upper P comma epsilon Baseline left-parenthesis lamda f 1 plus left-parenthesis 1 minus lamda right-parenthesis f 2 right-parenthesis period">

λ

V

P
,
δ

(

f
1

)
+
(
1
−
λ
)

V

P
,
δ

(

f
2

)
⊅

V

P
,
ε

(
λ

f
1

+
(
1
−
λ
)

f
2

)
.

\lambda {V_{P,\delta }}({f_1}) + (1 - \lambda ){V_{P,\delta }}({f_2})\not \supset {V_{P,\varepsilon }}(\lambda {f_1} + (1 - \lambda ){f_2}).

\]
Thus the "formula" that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda upper V Subscript p comma epsilon Baseline left-parenthesis f 1 right-parenthesis plus left-parenthesis 1 minus lamda right-parenthesis upper V Subscript p comma epsilon Baseline left-parenthesis f 2 right-parenthesis equals upper V Subscript p comma epsilon Baseline left-parenthesis lamda f 1 plus left-parenthesis 1 minus lamda right-parenthesis f 2 right-parenthesis">

λ

V

p
,
ε

(

f
1

)
+
(
1
−
λ
)

V

p
,
ε

(

f
2

)
=

V

p
,
ε

(
λ

f
1

+
(
1
−
λ
)

f
2

)

\lambda {V_{p,\varepsilon }}({f_1}) + (1 - \lambda ){V_{p,\varepsilon }}({f_2}) = {V_{p,\varepsilon }}(\lambda {f_1} + (1 - \lambda ){f_2})

given by Ghoussoub et al. in Mem. Amer. Math. Soc., vol. 70, no. 378, which is used there to establish the weak stability of F, is false.