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

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^{*} -algebra C^{*}(F_{R}) is isomorphic to the AF core of C^{*}(E_{R}) . Examples include C^{*} -algebras of quantum flag manifolds and quantum teardrops. We discuss in detail the quantum Grassmannian \operatorname{Gr}_{q}(2,4) and use our description as AF core to study its CW-structure.