MiS Preprint Repository

We prove that the graph C*-algebra $C^{\ast }(E)$ of a trimmable graph $E$ is $U(1)$-equivariantly isomorphic to a pullback C*-algebra of a subgraph C*-algebra $C^{\ast }(E^{″})$ and the C*-algebra of functions on a circle tensored with another subgraph C*-algebra $C^{\ast }(E^{\prime })$. This allows us to unravel the structure and K-theory of the fixed-point subalgebra $C^{\ast }(E{)}^{U(1)}$ through the (typically simpler) C*-algebras $C^{\ast }(E^{\prime })$, $C^{\ast }(E^{″})$ and $C^{\ast }(E^{″}{)}^{U(1)}$. As examples of trimmable graphs, we consider one-loop extensions of the standard graphs encoding respectively the Cuntz algebra ${\mathcal{O}}_2$ and the Toeplitz algebra $\mathcal{T}$. Then we analyze equivariant pullback structures of trimmable graphs yielding the C*-algebras of the Vaksman--Soibelman quantum sphere $S_q^{2n+1}$ and the quantum lens space $L_q^3(l;1,l)$, respectively.