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

2019/04/13 by Pere Ara, Ara, Pere, Roozbeh Hazrat +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1904.06468

openalex publication_date 2019/04/13 · openalex created_date 2019/04/25 · openalex updated_date 2026/07/28

Abstract

We prove that an isomorphism of graded Grothendieck groups K^gr₀ 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^gr₀(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^*-algebras. This result was only known for irreducible graphs.

Related

• The graded Grothendieck group and the classification of Leavitt path algebras
• The Graded Classification Conjecture holds for graphs with disjoint cycles
• Graded homotopy classification of Leavitt path algebras
• Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs
• The functor K₀^gr is full and only weakly faithful