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Invariant Banach limits and applications. (English) Zbl 1205.46012
Let \(\ell _{\infty }\) be the space of all bounded sequences \(x=(x_{1},x_{2},\ldots )\) with the norm \(\|x\|_{\ell_{\infty}} = \text{ sup}_n |x_n|\) and let \(L(\ell _{\infty })\) be the set of all bounded linear operators on \(\ell _{\infty }\). A linear functional \(B\) in the dual space \(\ell_\infty^\ast\) is said to be a Banach limit if \(B(1,1, \dots)=1\), \(B \geq 0\) and \(B(Tx)=B(x)\) for every \(x \in \ell_\infty\), where \(T\), is the translation operator, that is, \(T(x_1,x_2, \dots)=(x_2,x_3, \dots)\).
Following an approach similar to that of W. F. Eberlein for regular Hausdorff transformations in the classical paper [Proc. Am. Math. Soc. 1, 662–665 (1950; Zbl 0039.12102)] and motivated by their own recent contributions for the dilation operator \(\sigma_n\), \(n \in \mathbb{N}\), and the classical Cesàro operator \(C\), the authors present a set of easily verifiable sufficient conditions on an operator \(H \in L(\ell _{\infty })\), guaranteeing the existence of a Banach limit \(B\) on \(\ell _{\infty }\) such that \(B=BH\). They apply their results to the above mentioned Cesàro operator \(C\) on \(\ell _{\infty }\) and give a necessary and sufficient condition for an element \(x \in \ell _{\infty }\) to have fixed value \(Bx\) for all Cesàro invariant Banach limits \(B\). Finally, they apply the preceding description to obtain a characterization of “measurable elements” from the (Dixmier-)Macaev-Sargent ideal of compact operators with respect to an important subclass of Dixmier traces generated by all Cesàro-invariant Banach limits. It is shown that this class is strictly larger than the class of all “measurable elements” with respect to the class of all Dixmier traces.

MSC:
46B99 Normed linear spaces and Banach spaces; Banach lattices
46L51 Noncommutative measure and integration
40J05 Summability in abstract structures (should also be assigned at least one other classification number from Section 40-XX)
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