Dirac Operators on Quantum Projective Spaces

We construct a family of self-adjoint operators D _N , ${Nin{mathbb Z}}We construct a family of self-adjoint operators D _N , N ? mathbb Z{Nin{mathbb Z}} , which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space mathbb CP^l_q{{mathbb C}{rm P}^{ell}_q} , for any ℓ ≥ 2 and 0 < q < 1. They provide 0⁺-dimensional equivariant even spectral triples. If ℓ is odd and N=frac12(l+1){N=frac{1}{2}(ell+1)} , the spectral triple is real with KO-dimension 2ℓ mod 8.