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The weak stability of the positive face in $ L\sp 1$


Author: Zhibao Hu
Journal: Proc. Amer. Math. Soc. 123 (1995), 131-134
MSC: Primary 46E30; Secondary 46B20
DOI: https://doi.org/10.1090/S0002-9939-1995-1213862-3
MathSciNet review: 1213862
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Abstract: Let F be the positive face of the unit ball of $ {L^1}[0,1]$. We show that F is weakly stable in the sense that the midpoint map $ {\Phi _{1/2}}:F \times F \to F$, with $ {\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]$ to a Banach space X. We prove this stability by showing that if $ {f_1},{f_2} \in F,\lambda \in (0,1),\varepsilon > 0$ and $ \delta \geq \max \{ 2\varepsilon /\lambda ,2\varepsilon /(1 - \lambda )\} $, then

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

where $ P = \{ {A_1}, \ldots ,{A_n}\} $ is a finite positive partition of [0, 1] and

$\displaystyle {V_{P,\varepsilon }}(f) = \left\{ {g \in F:\sum\limits_{i = 1}^n ... ... {\int_{{A_i}} {(f - g)(t)d\mu (t)} } \right\vert \leq \varepsilon } } \right\}$

for any f in F. We construct an example showing that for any $ 0 < \lambda < 1$ there are functions $ {f_1}$ and $ {f_2}$ in F such that if $ 0 < \varepsilon < 2\min \{ \lambda ,1 - \lambda \} $ and $ 0 \leq \delta < \max \{ \varepsilon /\lambda ,\varepsilon /(1 - \lambda )\} $, then

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

Thus the "formula" that $ \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.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1213862-3
Keywords: Weakly stable, huskable operators, strongly regular operators, $ {L^1}[0,1]$
Article copyright: © Copyright 1995 American Mathematical Society

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