Extreme points of the set of Banach limits

Extreme points of the set of Banach limits Let $${\mathbb{N}}$$ be the set of all natural numbers and $${\ell_\infty=\ell_\infty (\mathbb{N})}$$ be the Banach space of all bounded sequences x = (x 1, x 2 . . .) with the norm $$\|x\|_{\infty}=\sup_{n\in\mathbb{N}}|x_n|,$$ and let $${\ell_\infty^*}$$ be its Banach dual. Let $${\mathfrak{B} \subset \ell_\infty^*}$$ be the set of all normalised positive translation invariant functionals (Banach limits) on ℓ ∞ and let $${ext(\mathfrak{B})}$$ be the set of all extreme points of $${\mathfrak{B}}$$ . We prove that an arbitrary sequence (B j ) j ≥ 1, of distinct points from the set $${ext(\mathfrak{B})}$$ is 1-equivalent to the unit vector basis of the space ℓ 1 of all summable sequences. We also study Cesáro-invariant Banach limits. In particular, we prove that the norm closed convex hull of $${ext(\mathfrak{B})}$$ does not contain a Cesáro-invariant Banach limit. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Extreme points of the set of Banach limits

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2012 by Springer Basel AG
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0155-3
Publisher site
See Article on Publisher Site

Abstract

Let $${\mathbb{N}}$$ be the set of all natural numbers and $${\ell_\infty=\ell_\infty (\mathbb{N})}$$ be the Banach space of all bounded sequences x = (x 1, x 2 . . .) with the norm $$\|x\|_{\infty}=\sup_{n\in\mathbb{N}}|x_n|,$$ and let $${\ell_\infty^*}$$ be its Banach dual. Let $${\mathfrak{B} \subset \ell_\infty^*}$$ be the set of all normalised positive translation invariant functionals (Banach limits) on ℓ ∞ and let $${ext(\mathfrak{B})}$$ be the set of all extreme points of $${\mathfrak{B}}$$ . We prove that an arbitrary sequence (B j ) j ≥ 1, of distinct points from the set $${ext(\mathfrak{B})}$$ is 1-equivalent to the unit vector basis of the space ℓ 1 of all summable sequences. We also study Cesáro-invariant Banach limits. In particular, we prove that the norm closed convex hull of $${ext(\mathfrak{B})}$$ does not contain a Cesáro-invariant Banach limit.

Journal

PositivitySpringer Journals

Published: Jan 17, 2012

References

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