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arXiv · 1904.06468

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

Pere Ara·Roozbeh Hazrat·Huanhuan Li
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Abstract

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, $K^{gr}_0\big(L(E)\big)$ 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 $C^*$-algebras. This result was only known for irreducible graphs.

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Pere Ara, Roozbeh Hazrat, Huanhuan Li. 2019-04-13. Graded $K$-Theory, Filtered $K$-theory and the classification of graph algebras. https://arxiv.org/abs/1904.06468

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