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Abstract
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We prove that an isomorphism of graded Grothendieck groups
of two
Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered
-theory
and consequently an isomorphism of filtered
-theory of their associated
graph
-algebras.
As an application, we show that since for a finite graph
with no sinks,
of the Leavitt
path algebra
coincides with Krieger’s dimension group of its adjacency matrix
,
our result relates the shift equivalence of graphs to the filtered
-theory and
consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph
-algebras.
This result was only known for irreducible graphs.
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Keywords
Leavitt path algebra, graph $C^*$-algebra, graded
$K$-theory, filtered $K$-theory, graded prime ideal, graded
Grothendieck group
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Mathematical Subject Classification
Primary: 16D70
Secondary: 18F30
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Milestones
Received: 2 February 2022
Revised: 21 July 2022
Accepted: 23 August 2022
Published: 15 March 2023
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