Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Banach limits and traces on $\mathcal L_{1,\infty}$   We introduce a new approach to traces on the principal ideal $\mathcal
L_{1,\infty}$ generated by any positive compact operator whose singular value
sequence is the harmonic sequence. Distinct from the well-known construction of
J.~Dixmier, the new approach provides the explicit construction of every trace
of every operator in $\mathcal L_{1,\infty}$ in terms of translation invariant
functionals applied to a sequence of restricted sums of eigenvalues. The
approach is based on a remarkable bijection between the set of all traces on
$\mathcal L_{1,\infty}$ and the set of all translation invariant functionals on
$l_\infty$. This bijection allows us to identify all known and commonly used
subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of
invariance properties of linear functionals on $l_\infty$, and definitively
classify the measurability of operators in $\mathcal L_{1,\infty}$ in terms of
qualified convergence of sums of eigenvalues. This classification has led us to
a resolution of several open problems (for the class $\mathcal L_{1,\infty}$)
from~\cite{CS}. As an application we extend Connes' classical trace theorem to
positive normalised traces.

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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph