On amplified graph $C^{\ast }$-algebras as cores of Cuntz–Krieger algebras

Abstract

Given a finite directed acyclic graph $R$, we construct from it two graphs $E_R$ and $F_R$, one by adding a loop at every vertex of $R$ and one by replacing every arrow of $R$ by countably infinitely many arrows. We show that the graph $C^{\ast }$-algebra $C^{\ast }(F_R)$ is isomorphic to the AF core of $C^{\ast }(E_R)$. Examples include $C^{\ast }$-algebras of quantum flag manifolds and quantum teardrops. We discuss in detail the quantum Grassmannian ${\mathrm{Gr}}_q(2,4)$ and use our description as AF core to study its CW-structure.