Real dimensional spaces in noncommutative geometry

In this paper we will extend the product of spectral triples to a product of semifinite spectral triples. We will prove that finite summability and regularity are preserved under taking products. Connes and Marcolli constructed for each z∈(0,∞)z∈(0,∞) a type II∞II∞-semifinite spectral triple which can be considered as a geometric space of dimension z. A small adaption of their construction yields a type I-semifinite spectral triple. The properties of these semifinite spectral triples will be investigated. At the same time we will also avoid the need for an infra-red cutoff to compute the dimension spectrum. An application of these semifinite spectral triples to dimensional regularization in quantum field theory is given.