# Quantum complex projective spaces from Toeplitz cubes

### Piotr M. Hajac

IMPAN, Warsaw, Poland### Atabey Kaygun

Bahçeşehir University, Istanbul, Turkey### Bartosz Zieliński

University of Lodz, Poland

## 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.

## Cite this article

Piotr M. Hajac, Atabey Kaygun, Bartosz Zieliński, Quantum complex projective spaces from Toeplitz cubes. J. Noncommut. Geom. 6 (2012), no. 3, pp. 603–621

DOI 10.4171/JNCG/100