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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)
##### Keywords:
Cesàro-invariant Banach limits; singular traces
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##### References:
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