Diagonal-preserving Isomorphisms of Algebras from Infinite Graphs

We establish logical equivalence between statements involving the Cuntz [Formula: see text]-algebra [Formula: see text] with its canonical diagonal; graph [Formula: see text]-algebras with their canonical diagonals; Leavitt path algebras over general fields with their canonical diagonals; Leavitt path algebras over [Formula: see text]; topological full groups; groupoids; and the automorphism [Formula: see text] on certain [Formula: see text]- and homology groups equal to [Formula: see text]. Deciding whether these equivalent statements are true or false is of importance in studies of geometric classification of diagonal-preserving isomorphism between graph [Formula: see text]-algebras and Leavitt path algebras, mirroring a similar hindrance studied by Cuntz more than 40 years ago.