Injective strong commutativity preservers on \(\mathcal{T}_{\infty}(F)\) (Q520005)

In the introduction, the authors define a commutativity preserver, preservers of commutativity in both directions, and a strong commutativity preserver (Lie homeomorphism). The authors study these three types of maps on the algebra of infinite upper triangular matrices \(\mathcal{T}_{\infty}(F)\) over a field \(F\) such that \(\text{char}(F)\neq 2\). For any invertible triangular matrix \(t\) from \(\mathcal{T}_{\infty}(F)\), there are definitions of the inner automorphism \(\mathcal{I}nn_t(x) = t^{-1}xt\), another mapping \(\mathcal{S}pl_{\tau} (x)\) for an injection \(\tau: \mathbb{N}\rightarrow \mathbb{N}\), and also a little bit complicated mapping \(\mathcal{C}ut_N\). All these definitions are illustrated with examples. The main result of the paper gives a description of injective strong commutativity preservers on \(\mathcal{T}_{\infty}(F)\). Another theorem presents some results concerning the maps that preserve commutativity in both directions on \(\mathcal{T}_{\infty}(F)\). Finally, in two theorems in Section 3, the authors give conditions for a mapping \(\phi: \mathcal{T}_{\infty}(F)\rightarrow \mathcal{T}_{\infty}(F)\) to be a separable sum of maps.