The Graded Classification Conjecture holds for graphs with disjoint cycles

Abstract

The Graded Classification Conjecture (GCC) states that the pointed $K_0^{\operatorname{gr}}$-group is a complete invariant of the Leavitt path algebras of finite graphs when these algebras are considered with their natural grading by $\mathbb Z.$ The conjecture has previously been shown to hold in some special cases. The main result of the paper shows that the GCC holds for a significantly more general class of graphs included in the class of graphs with disjoint cycles. In particular, our result holds for finite graphs with disjoint cycles. We show the main result also for graph $C^*$-algebras. As a consequence, the graded version of the Isomorphism Conjecture holds for the class of graphs we consider. Besides showing the conjecture for the class of graphs we consider, we realize the Grothendieck $\mathbb Z$-group isomorphism by a specific graded $*$-isomorphism. In particular, we introduce a series of graph operations which preserve the graded $*$-isomorphism class of their algebras. After performing these operations on a graph, we obtain well-behaved ``representative'' graphs, which we call canonical forms. We define an equivalence $\approx$ on graphs such that $E\approx F$ holds when there are isomorphic canonical forms of $E$ and $F$ and we show that the condition $E\approx F$ is equivalent to the existence of an isomorphism $f$ of the Grothendieck $\mathbb Z$-groups of the algebras of $E$ and $F$ in the appropriate category. As $E\approx F$ can be realized by a finite series of specific graph operations, any such isomorphism $f$ can be realized by an explicit graded $*$-algebra isomorphism. Thus, we describe the graded ($*$-)isomorphism classes of the algebras of graphs we consider. Besides the ties to symbolic dynamics and Williams' Problem, such a description is relevant for the active program of classification of graph $C^*$-algebras.