21 research outputs found
Remarks on Morphisms of Spectral Geometries
Having in view the study of a version of Gel'fand-Neumark duality adapted to
the context of Alain Connes' spectral triples, in this very preliminary review,
we first present a description of the relevant categories of geometrical
spaces, namely compact Hausdorff smooth finite-dimensional orientable
Riemannian manifolds (or more generally Hermitian bundles of Clifford modules
over them); we give some tentative definitions of the relevant categories of
algebraic structures, namely "propagators" and "spectral correspondences" of
commutative Riemannian spectral triples; and we provide a construction of
functors that associate a naive morphism of spectral triples to every smooth
(totally geodesic) map. The full construction of spectrum functors
(reconstruction theorem for morphisms) and a proof of duality between the
previous "geometrical' and "algebraic" categories are postponed to subsequent
works, but we provide here some hints in this direction. We also show how the
previous categories of "propagators" of commutative C*-algebras embed in the
mildly non-commutative environments of categories of suitable Hilbert
C*-bimodules, factorizable over commutative C*-algebras, with composition given
by internal tensor product.Comment: 9 pages, AMS-LaTeX2e. Reformatted, heavily revised and corrected
version, only for arXiv, of a previous review paper published in East-West
Journal of Mathematics. The main results presented in this review are now
part of F.Jaffrennou PhD thesis "Morphisms of Spectral Geometries" (Mahidol
University, June 2014
Enriched Fell Bundles and Spaceoids
We propose a definition of involutive categorical bundle (Fell bundle)
enriched in an involutive monoidal category and we argue that such a structure
is a possible suitable environment for the formalization of different
equivalent versions of spectral data for commutative C*-categories.Comment: 12 pages, AMS-LaTeX2e, to be published in "Proceedings of 2010 RIMS
Thematic Year on Perspectives in Deformation Quantization and Noncommutative
Geometry
Modular Theory, Non-Commutative Geometry and Quantum Gravity
This paper contains the first written exposition of some ideas (announced in
a previous survey) on an approach to quantum gravity based on Tomita-Takesaki
modular theory and A. Connes non-commutative geometry aiming at the
reconstruction of spectral geometries from an operational formalism of states
and categories of observables in a covariant theory. Care has been taken to
provide a coverage of the relevant background on modular theory, its
applications in non-commutative geometry and physics and to the detailed
discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields
A Category of Spectral Triples and Discrete Groups with Length Function
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.Comment: 23 pages, AMS-LaTeX2
A Remark on Gelfand Duality for Spectral Triples
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.Comment: 15 pages, AMS-LaTeX2e, results unchanged, several improvements in the
exposition, appendix adde