Graded Classification Conjecture for Leavitt path algebras

Establish that the pointed pre-ordered Z[x, x^{-1}]-module K_0^{gr}(L_k(E)) completely classifies Leavitt path algebras (and analogously graph C*-algebras), i.e., prove that for directed graphs E and F the existence of an order-preserving isomorphism of their graded Grothendieck groups implies, and is implied by, a graded isomorphism of the Leavitt path algebras L_k(E) and L_k(F).

Background

The paper surveys how graded K-theory functions as a classification tool for graph algebras arising from symbolic dynamics. A central theme is the (still unresolved) Graded Classification Conjecture asserting that the graded Grothendieck group K_0^{gr}, viewed as a pointed pre-ordered Z[x, x^{-1}]-module, is a complete invariant for Leavitt path algebras (and similarly for graph C*-algebras).

Establishing this conjecture would unify algebraic and dynamical classifications; for example, it would imply that certain dynamical equivalences (e.g., eventual conjugacy or shift equivalence of shifts of finite type) correspond precisely to graded Morita equivalence of the associated Leavitt path algebras. Considerable progress exists on fullness and related properties, but a complete classification via K_0^{gr} is not yet proven.