Abstract

We prove that an isomorphism of graded Grothendieck groups $K_0^{gr}$ of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered $K$-theory and consequently an isomorphism of filtered $K$-theory of their associated graph $C^{\ast }$-algebras. As an application, we show that since for a finite graph $E$ with no sinks, $K_0^{gr}(L(E))$ of the Leavitt path algebra $L(E)$ coincides with Krieger’s dimension group of its adjacency matrix $A_E$, our result relates the shift equivalence of graphs to the filtered $K$-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph $C^{\ast }$-algebras. This result was only known for irreducible graphs.

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