Limit algebras of differential forms in non-commutative geometry

Abstract

Given a C^*-normed algebra A which is either a Banach *-algebra or a Frechet *-algebra, we study the algebras Ω_∞ A and Ω_∈ A obtained by taking respectively the projective limit and the inductive limit of Banach *-algebras obtained by completing the universal graded differential algebra Ω^*A of abstract non-commutative differential forms over A. Various quantized integrals on Ω_∞A induced by a K-cycle on A are considered. The GNS-representation of Ω_∞A defined by a d-dimensional non-commutative volume integral on a d ⁺-summable K-cycle on A is realized as the representation induced by the left action of A on Ω^*A. This supplements the representation A on the space of forms discussed by Connes.