The Cuntz splice does not preserve *-isomorphism of Leavitt path algebras over ℤ

2015/07/05 by Rune Johansen, Johansen, Rune, Adam P. W. Sørensen +1
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1507.01247

openalex publication_date 2015/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the Leavitt path algebras L_2,ℤ and L_2-,ℤ are not isomorphic as *-algebras. There are two key ingredients in the proof. One is a partial algebraic translation of Matsumoto and Matui's result on diagonal preserving isomorphisms of Cuntz--Krieger algebras. The other is a complete description of the projections in L_ℤ(E) for E a finite graph. This description is based on a generalization, due to Chris Smith, of the description of the unitaries in L_2,ℤ given by Brownlowe and the second named author. The techniques generalize to a slightly larger class of rings than just ℤ.

Related

• Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs
• Distinguishing Leavitt algebras among Leavitt path algebras of finite graphs by Serre property
• Algebraic Cuntz-Krieger algebras
• The Leavitt path algebra of a graph
• A note on the isomorphism conjectures for Leavitt path algebras