An equivariant pullback structure of trimmable graph C*-algebras

We introduce a class of graphs called trimmable. Then we show that the Leavitt path algebra of a trimmable graph is graded-isomorphic to a pullback algebra of simpler Leavitt path algebras and their tensor products.

Next, specializing the ground field to the field of complex numbers and completing Leavitt path algebras to graph C*-algbras, we prove that the graph C*-algebra of a trimmable graph is U(1)-equivariantly isomorphic with an appropriate pullback C*-algebra.

As a main application, we consider a trimmable graph yielding the C*-algebra $C(S^{2n+1}_q)$ of the Vaksman-Soibelman quantum sphere, and use the resulting pullback structure of its gauge invariant subalgebra $C(CP^n_q)$ defining the quantum complex projective space to show that the generators of the even K-group of $C(CP^n_q)$ are given by a Milnor connecting homomorphism applied to the (unique up to sign) generator of the odd K-group of $C(S^{2n-1}_q)$ and by the generators of the even K-group of $C(CP^{n-1}_q)$.

Based on joint works with Francesco D'Andrea, Atabey Kaygun and Mariusz Tobolski.