Fuzzy de Sitter Space

We discuss properties of fuzzy de Sitter space defined by means of algebra of the de Sitter group $\mathrm{SO}(1,4)$ in unitary irreducible representations. It was shown before that this fuzzy space has local frames with metrics that reduce, in the commutative limit, to the de Sitter metric. Here we determine spectra of the embedding coordinates for $(\rho,s=\frac 12)$ unitary irreducible representations of the principal continuous series of the $\mathrm{SO}(1,4)$. The result is obtained in the Hilbert space representation, but using representation theory it can be generalized to all representations of the principal continuous series.