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Register a new userA Remark on Gelfand Duality for Spectral Triples
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We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable metric category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a quotient of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.
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In the setting of C*-categories, we provide a definition of spectrum of a commutative full C*-category as a one-dimensional unital saturated Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gelfand duality theorem generalizing the usual Gelfand duality between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we glue them together seems to shed some new light on the subject.
In the context of A. Connes spectral triples, a suitable notion of morphism is introduced. Discrete groups with length function provide a natural example for our definitions. A. Connes construction of spectral triples for group algebras is a covariant functor from the category of discrete groups with length functions to that of spectral triples. Several interesting lines for future study of the categorical properties of spectral triples and their variants are suggested.
We present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras.
After recalling in detail some basic definitions on Hilbert C*-bimodules, Morita equivalence and imprimitivity, we discuss a spectral reconstruction theorem for imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras and consider some of its applications in the theory of commutative full C*-categories.
In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $Lambda$, via the infinite path space $Lambda^infty$ of $Lambda$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $Lambda^infty$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph $Lambda$. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are $zeta$-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure $mu$, and show that $mu$ is a rescaled version of the measure $M$ on $Lambda^infty$ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of $L^2(Lambda^infty, M)$ which was constructed by Farsi et al.
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