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

Pere Ara, Roozbeh Hazrat and Huanhuan Li

Vol. 7 (2022), No. 4, 731–795
DOI: 10.2140/akt.2022.7.731
Abstract

We prove that an isomorphism of graded Grothendieck groups K0gr 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, K0gr(L(E)) of the Leavitt path algebra L(E) coincides with Krieger’s dimension group of its adjacency matrix AE, 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.

Keywords
Leavitt path algebra, graph $C^*$-algebra, graded $K$-theory, filtered $K$-theory, graded prime ideal, graded Grothendieck group
Mathematical Subject Classification
Primary: 16D70
Secondary: 18F30
Milestones
Received: 2 February 2022
Revised: 21 July 2022
Accepted: 23 August 2022
Published: 15 March 2023
Authors
Pere Ara
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Bellaterra
Spain
Barcelona Graduate School of Mathematics
Barcelona
Spain
Roozbeh Hazrat
Centre for Research in Mathematics and Data Science
Western Sydney University
Sydney, NSW
Australia
Huanhuan Li
Center for Pure Mathematics
School of Mathematical Sicences
Anhui University
Hefei, Anhui
China