Cell complex structure of quantum projective spaces

I will explain how the graph C*-algebras of a trimmable graph can be decomposed as U(1)-equivariant pullback of two simpler C*-algebras. A main example is given by the algebra of Vaksman-Soibelman quantum sphere, that can be realized as pushout of a lower dimensional quantum sphere and the product of a quantum ball with a circle (we understand the pullback of C*-algebras as pushout of the underlying "noncommutative spaces"). The U(1)-invariant part of this pullback diagram gives a "CW complex" realization of quantum projective spaces that allows to give an explicit description of the K-theory generators. Further examples include quantum lens spaces, one-loop extensions of Cuntz algebras of the Toeplitz algebra. This is a joint work with Francesca Arici, Piotr M. Hajac and Mariusz Tobolski.