Under Weyl-type eigenvalue asymptotics, the normalized truncated-trace functional equals the Dixmier-trace noncommutative integral after logarithmic averaging, yielding Szegő limit formulas, a density-of-states theorem, and a quantum-ergodicity-style theorem for spectral triples.
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A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity
Under Weyl-type eigenvalue asymptotics, the normalized truncated-trace functional equals the Dixmier-trace noncommutative integral after logarithmic averaging, yielding Szegő limit formulas, a density-of-states theorem, and a quantum-ergodicity-style theorem for spectral triples.