Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs

TL;DR

This paper explores the logical equivalences among various algebraic and topological structures related to infinite graphs, aiming to address classification challenges in diagonal-preserving isomorphisms of graph C*-algebras and Leavitt path algebras.

Contribution

It establishes a framework linking multiple algebraic and topological invariants, facilitating the classification of diagonal-preserving isomorphisms in graph-related algebras.

Findings

Logical equivalences among algebraic and topological structures

Connection between automorphisms and K-theory groups

Implications for geometric classification of graph algebras

Abstract

We establish logical equivalence between statements involving * the Cuntz C*-algebra $\mathcal O_\infty$ 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 $x\mapsto -x$ on certain $K_0$- 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.