Examples of non integer dimensional geometries
  Two examples of spectral triples with non-integer dimension spectrum are considered. These triples involve commutative C*-algebras. The first example has complex dimension spectrum and trivial differential algebra. The other is a parameter dependent deformation of the canonical spectral triple over S1. Its dimension spectrum includes real non-integer values. It has a non-trivial differential algebra and in contrast with the one dimensional case there are no junk forms for a non-vanishing deformation parameter. The distance on this space depends non-trivially on this parameter. 
R. Trinchero