Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs
Søren Eilers, Efren Ruiz
TL;DR
The paper studies diagonal-preserving isomorphisms among algebras arising from infinite graphs, linking $\mathcal{O}_\infty$, graph $C^*$-algebras, Leavitt path algebras, and their diagonals to topological full groups and groupoids through $K_0$-theory and zeroth groupoid homology. It extends Matsumoto's diagonal-normalizer framework to the infinite-graph setting by introducing diagonal-normalizer monoids $\mathcal{V}^*(A,D)$ and proving isomorphisms with the graph monoid $M_E$, across $\mathbb{Z}$, general fields, and $\mathbb{C}$-coefficients, via Steinberg algebras and groupoid techniques. A central result is that several diagonal-preserving isomorphism questions for $\mathcal{O}_\infty$, $C^*(E)$, $L_\mathsf{k}(E)$, and related groupoids are equivalent, and that solving these reduces to the existence of diagonal-preserving stable automorphisms inducing the nontrivial $K_0$-automorphism (or corresponding homology automorphism). The work provides a unifying framework—bridging operator algebras, graph algebras, and groupoid methods—that advances the classification program for diagonal-preserving isomorphisms in the infinite-graph regime and informs rigidity phenomena across these categories.
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.