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 cate

gories of geometrical spaces, namely compact Hausdorff smooth finite

dimensional orientable Riemannian manifolds, or more generally Hermi

tian 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 mor

phism of spectral triples to every smooth (totally geodesic) map. The

full construction of spectrum functors (reconstruction theorem for mor

phisms) 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 de

scription of morphisms in non-commutative geometry