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Studia Mathematica

2013 | 214 | 1 | 37-66
Tytuł artykułu

Shift-invariant functionals on Banach sequence spaces

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The present paper is a continuation of [23], from which we know that the theory of traces on the Marcinkiewicz operator ideal
$𝔐 (H):= {T ∈ 𝔏(H): sup_{1≤m<∞} 1/(log m + 1) ∑_{n=1}^{m} aₙ(T) < ∞}$
can be reduced to the theory of shift-invariant functionals on the Banach sequence space
$𝔴(ℕ₀):= {c = (γ_{l}): sup_{0≤k<∞} 1/(k+1) ∑_{l=0}^{k} |γ_{l}| < ∞}$.

The final purpose of my studies, which will be finished in [24], is the following. Using the density character as a measure, I want to determine the size of some subspaces of the dual 𝔐 *(H). Of particular interest are the sets formed by the Dixmier traces and the Connes-Dixmier traces (see [2], [4], [6], and [13]).
As an intermediate step, the corresponding subspaces of 𝔴*(ℕ₀) are treated. This approach has a significant advantage, since non-commutative problems turn into commutative ones.
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Tom
Numer
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37-66
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wydano
2013
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autor
• Biberweg 7, D-07749 Jena, Germany
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