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Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebras

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 نشر من قبل Roozbeh Hazrat
 تاريخ النشر 2019
  مجال البحث
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We prove that an isomorphism of graded Grothendieck groups $K^{gr}_0$ 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 $C^*$-algebras. As an application, we show that, since for a finite graph $E$ with no sinks, $K^{gr}_0big(L(E)big)$ of the Leavitt path algebra $L(E)$ coincides with Kriegers 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 $C^*$-algebras. This result was only known for irreducible graphs.



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