An equivariant pullback structure of trimmable graph $C^{\ast }$-algebras

Abstract

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