unital involutive associative algebra and by a representation of a $*$-algebra we always mean a unital $*$-representation. We will denote by ${\underline}{m}=(m_0,m_1,\ldots,m_n)$ a vector with $n+1$ components (labelled from $0$ to $n$), and by $\vec{m}=(m_1,\ldots,m_n)$ a vector with $n$ components (labelled from $1$ to $n$), when $n\geq 1$ is the complex dimension of the space considered.

*Acknowledgments.* We are grateful to A. De Paris, M. Frank and T. Brzezi[ń]{}ski for useful discussions and correspondences. Part of the work was done at the Hausdorff Research Institute for Mathematics in Bonn during the 2014 Trimester Program “Non-commutative Geometry and its Applications". We thank the organizers of the Program for the invitation and all people at HIM for the nice hospitality.

Weighted projective spaces {#sec:tre}
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Classical weighted projective spaces ${\mathbb{P}}({\underline}\ell)$ are among the best known examples of projective toric varieties. As quotient spaces they have a natural orbifold structure and include, in complex dimension 1, the orbifolds named teardrops by Thurston in [@Thu80]. We start by recalling some basic facts about their classification as done in [@BFNR13].

A *weight vector* ${\underline}{\ell}=(\ell_0,\ldots,\ell_n)$ is a finite sequence of positive integers, called *weights*. A weight vector is *normalized* if for any prime $p$ at least 