Spectral triples for AF C*-algebras and metrics on the cantor set
By Cristina Antonescu, Erik Christensen. 2003
103 citations recorded by the source corpus.
An AF C -algebra has a natural filtration as an increasing se- quence of finite dimensional C -algebras. We show that it is possible to con- struct a Dirac operator which relates to this filtration in a natural way and which will induce a metric for the weak*-topology on the state space of the algebra. It turns out that for AF C -algebras, there is no limit to the growth of the eigenvalues of such a Dirac operator. We have obtained a kind of an in- verse to this result, by showing that a phenomenon like this can only occur for AF C -algebras. The results are then applied to a study of the classical Cantor set.
Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/8433bc71b652b9aad1f5db8699ebd9be960608d4
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In Algebra and Number Theory
In Mathematical Physics
- J. Bellissard
- Kamran Reihani
- Matilde Marcolli
- Michel L. Lapidus
- Erik Christensen
- Cristina Ivan
- Joseph Keir
- Yuri I. Manin
In Statistical and Nonlinear Physics
- Matilde Marcolli
- J. Bellissard
- Kamran Reihani
- Michel L. Lapidus
- Erik Christensen
- Cristina Ivan
- Joseph Keir
- Jeffrey C. Lagarias
Citations in the corpus
- Dirac operators and spectral triples for some fractal sets built on curves (2006, background): “In [2], Erik Christensen and Cristina Ivan constructed a spectral triple for the approximately finite-dimensional (AF) C*-algebras.”
- Quantum Ultrametrics on AF Algebras and The Gromov-Hausdorff Propinquity (2015, unknown): “We should note that for any two states φ,ψ in the state space S (A) of a unital C*-algebra A, the function t ∈ [0, 1] 7→ tφ+(1−t)ψ is a continuous function to S (A) equipped with the weak* topology.”
- Dynamical Systems on Spectral Metric Spaces (2010, methodology): “Rennie and Varilly [59] and Connes [23]. Several important classes of noncommutative spectral triples have been constructed by Rieel [63] and by Christensen, Antonescu-Ivan and various collaborators [13, 14, 15, 16]. Recently, the Riemannian geometry of ultrametric Cantor sets was described in [55] ”