Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs

2025/11/11 by Eilers, Søren, Ruiz, Efren
#FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2511.07917

Abstract

We establish logical equivalence between statements involving * the Cuntz C*-algebra \mathcal O_∞ with its canonical diagonal; * graph C*-algebras with their canonical diagonals; * Leavitt path algebras over general fields with their canonical diagonals; * Leavitt path algebras over \mathbb Z; * topological full groups; * groupoids; and * the automorphism x↦ -x on certain K₀- and homology groups equal to \mathbb Z Deciding whether these equivalent statements are true or false is of importance in studies of geometric classification of diagonal-preserving isomorphism between graph C*-algebras and Leavitt path algebras, mirroring a similar hindrance studied by Cuntz more than 40 years ago.

Citations

• The Algebraic Kirchberg-Phillips Question for Leavitt path algebras
• A note on projections in étale groupoid algebras and diagonal preserving homomorphisms
• Geometric classification of isomorphism of unital graph C*-algebras
• Refined moves for structure-preserving isomorphism of graph C*-algebras
• Homology of étale groupoids, a graded approach
• Diagonal-preserving isomorphisms of étale groupoid algebras
• Equivalent groupoids have Morita equivalent Steinberg algebras
• Non-stable K-theory for Leavitt path algebras
• Classification of unital simple Leavitt path algebras of infinite graphs
• Geometric classification of simple graph algebras
• Leavitt path algebras of separated graphs
• Flow invariants in the classification of Leavitt path algebras
• Isomorphisms between Leavitt algebras and their matrix rings

Related

• Diagonal-preserving ring *-isomorphisms of Leavitt path algebras
• The Cuntz splice does not preserve *-isomorphism of Leavitt path algebras over ℤ
• Isomorphism and Morita equivalence of graph algebras
• A dynamical characterization of diagonal preserving *-isomorphisms of graph C^*-algebras
• Graded K-Theory, Filtered K-theory and the classification of graph algebras