Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs

We establish logical equivalence between statements involving * the Cuntz C*-algebra ${\mathcal{O}}_{\mathrm{\infty }}$ 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.