  In this note we introduce a new family of non-commutative spaces that we call
non-commutative toric varieties and we describe some of their main properties.
The main technical tool in this investigation is a natural extension of
LVM-theory for the irrational case. In order to introduce the moduli space of
(non-commutative) toric varieties we use variations on the notion of diffeology
as models for non-commutative spaces.
