General Preservers of Quasi-Commutativity

Abstract

Abstract. Let Mn be the algebra of all n × n matrices over C. We say that A, B ∈ Mn quasi-commute if there exists a nonzero ξ ∈ C such that AB = ξBA. In the paper we classify bijective not necessarily linear maps Φ : Mn → Mn which preserve quasi-commutativity in both directions.

On maximal distances in a commuting graph

“…Let S 1 be an invertible matrix such that Y = S Let us write Z = Z ij 1≤i,j≤3 and Z = Z ij 1≤i,j≤3 as 3 × 3 block matrices and by (7) we have that Z ij , Z ij ∈ Z 2 [C] = GF (2 3 ) ⊆ M 3 (Z 2 ), hence each of them is either zero or invertible. Then (8) implies Observe that each block on the left side belongs to Z 2 [C] = GF (2 3 ) ⊆ M 3 (Z 2 ), and so is either zero or invertible. On the other hand, on the right side, each block in the last two columns has rank at most two.…”

“…The other relation is a preorder given by A ≺ B if C(A) ⊆ C(B). It was already observed that minimal and maximal matrices in this poset are of special importance, see for example [7,20,8]. Recall that a matrix A is minimal if C(X) ⊆ C(A) implies C(X) = C(A).…”

General preservers of quasi-commutativity on self-adjoint operators

“…In our recent paper [4] we classified nonlinear bijective preservers of quasi-commutativity in both directions on the algebra of n × n complex matrices. Since in quantum mechanics self-adjoint operators are important we classified such maps also on the space of n × n hermitian matrices [5].…”