REMARKS ON MORPHISMS OF SPECTRAL GEOMETRIES

Abstract

Non-commutative geometry, conceived by Alain Connes, is a new branch of mathematics whose aim is the study of geometrical spaces using tools from operator algebras and functional analysis. Specifically metrics for non-commutative manifolds are now encoded via spectral triples, a set of data involving a Hilbert space, an algebra of operators acting on it and an unbounded self-adjoint operator, maybe endowed with supplemental structures. Our main objective is to prove a version of Gel’fand-Na˘ımark duality adapted to the context of Alain Connes’ spectral triples. In this preliminary exposition, we present: a description of the relevant categories of geometrical spaces, namely compact Hausdorff smooth finitedimensional orientable Riemannian manifolds, or more generally Hermitian bundles of Clifford modules over them; some tentative definitions of categories of algebraic structures, namely commutative Riemannian spectral triples; 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 conjecture how the previous “algebraic” categories might provide a suitable environment for the description of morphisms in non-commutative geometry.