Comparison between two differential graded algebras in Noncommutative Geometry

Abstract

Starting with a spectral triple one can associate two canonical differential graded algebras (dga) defined by Connes and Fr\"ohlich et al. For the classical spectral triples associated with compact Riemannian spin manifolds both these dgas coincide with the de-Rham dga. Therefore, both are candidates for the noncommutative space of differential forms. Here we compare these two dgas and observe that in a very precise sense Connes' dga is more informative than that of Fr\"ohlich et al.