Sums of two-dimensional spectral triples

Erik Christensen; Cristina Ivan

Sums of two-dimensional spectral triples

Institut for Matematiske Fag

18 Citationer (Scopus)

Abstract

Arbejdet studerer summer af 2 dimensionale moduler for algebraen af kontinuerte komplekse funktioner på et kompakt metrisk rum, og det vises at konstruktionen kan laves således, at den givne metrik kan rekonstrueres ud fra et spektralt triple der fremkommer som en sådan uendelig sum. Det vises at det spektrale triple også kan give estimater vedrørende det kompakte rums Minkowski dimension, og endelig anvendes konstruktionen på enhedsintervallet, hvor det vises at både metirk, dimension og Lebesguemål kan konstrueres ud fra dette spektrale triple.

Originalsprog	Engelsk
Tidsskrift	Mathematica Scandinavica
Vol/bind	100
Udgave nummer	1
Sider (fra-til)	35-60
Antal sider	26
ISSN	0025-5521
Status	Udgivet - 2007

Emneord

Det Natur- og Biovidenskabelige Fakultet
matematik
ikke kommutativ geometri

Citationsformater

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AU - Christensen, Erik

AU - Ivan, Cristina

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N2 - We study countable sums of two dimensional modules for the continuous complex functions on a compact metric space and show that it is possible to construct a spectral triple which gives the original metric back. This spectral triple will be finitely summable for any positive parameter. We also construct a sum of two dimensional modules which reflects some aspects of the topological dimensions of the compact metric space, but this will only give the metric back approximately. At the end we make an explicit computation of the last module for the unit interval in. The metric is recovered exactly, the Dixmier trace induces a multiple of the Lebesgue integral but the growth of the number of eigenvalues is different from the one found for the standard differential operator on the unit interval.

AB - We study countable sums of two dimensional modules for the continuous complex functions on a compact metric space and show that it is possible to construct a spectral triple which gives the original metric back. This spectral triple will be finitely summable for any positive parameter. We also construct a sum of two dimensional modules which reflects some aspects of the topological dimensions of the compact metric space, but this will only give the metric back approximately. At the end we make an explicit computation of the last module for the unit interval in. The metric is recovered exactly, the Dixmier trace induces a multiple of the Lebesgue integral but the growth of the number of eigenvalues is different from the one found for the standard differential operator on the unit interval.

KW - Faculty of Science

KW - matematik

KW - ikke kommutativ geometri

KW - mathematics

KW - non commutative geometry

M3 - Journal article

SN - 0025-5521

VL - 100

SP - 35

EP - 60

JO - Mathematica Scandinavica

JF - Mathematica Scandinavica

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ER -

Sums of two-dimensional spectral triples

Abstract

Emneord

Fingeraftryk

Citationsformater