Quantum complex projective spaces from Toeplitz cubes

Abstract

From $N$-tensor powers of the Toeplitz algebra, we construct a multi-pullback C*-algebra that is a noncommutative deformation of the complex projective space ${\mathbb{P}}^N(\mathbb{C})$. Using Birkhoff’s Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multi-pullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space.