An equivariant pullback structure of trimmable graph $C^*$-algebras

To unravel the structure of fundamental examples studied in noncommutative topology, we prove that graph $C^$-algebra $C^(E)$ a trimmable $E$ is $U(1)$-equivariantly isomorphic to pullback subgraph $C^(E'')$ and functions on circle tensored with another $C^(E')$. This allows us approach K-theory fixed-point subalgebra $C^(E)^{U(1)}$ through (typically simpler) $C^$-algebras $C^(E')$, $C^(E'')^{U(1)}$. As graphs, consider one-loop extensions standard graphs encoding respectively Cuntz algebra $\mathcal{O}\_2$ Toeplitz $\mathcal{T}$. Then analyze equivariant structures yielding Vaksman–Soibelman quantum sphere $S^{2n+1}\_q$ lens space $L\_q^3(l;1,l)$, respectively.