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

Abstract

To unravel the structure of fundamental examples studied in noncommutative topology, we prove that the graph $C^$-algebra $C^(E)$ of a trimmable graph $E$ is $U(1)$-equivariantly isomorphic to a pullback $C^$-algebra of a subgraph $C^$-algebra $C^(E'')$ and the $C^$-algebra of functions on a circle tensored with another subgraph $C^$-algebra $C^(E')$. This allows us to approach the structure and K-theory of the fixed-point subalgebra $C^(E)^{U(1)}$ through the (typically simpler) $C^$-algebras $C^(E')$, $C^(E'')$ and $C^(E'')^{U(1)}$. 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 $S^{2n+1}\_q$ and the quantum lens space $L\_q^3(l;1,l)$, respectively.