Suppose that $𝔽$ is a field and $m,n\ge \mathrm{3}$ are integers. Denote by $M_{mn}(𝔽)$ the set of all $m\times n$ matrices over $𝔽$ and by $M_n(𝔽)$ the set $M_{nn}(𝔽)$. Let $f_{ij}$ ($i\in [\mathrm{1},m],j\in [\mathrm{1},n]$) be functions on $𝔽$, where $[\mathrm{1},n]$ stands for the set $\{\mathrm{1},\dots ,n\}$. We say that a map $f:M_{mn}(𝔽)\to M_{mn}(𝔽)$ is induced by $\{f_{ij}\}$ if $f$ is defined by $f:[a_{ij}]\mapsto [f_{ij}(a_{ij})]$. We say that a map $f$ on $M_n(𝔽)$ preserves similarity if $A~B\Rightarrow f(A)~f(B)$, where $A~B$ represents that $A$ and $B$ are similar. A map $f$ on $M_n(𝔽)$ preserving inverses of matrices means $f(A)f(A^{-\mathrm{1}})=I_n$ for every invertible $A\in M_n(𝔽)$. In this paper, we characterize induced maps preserving similarity and inverses of matrices, respectively.

Suppose $m,n\ge \mathrm{2}$ are positive integers. Let ${𝒯}_n$ be the space of all $n\times n$ complex upper triangular matrices, and let $\varphi$ be an injective linear map on ${𝒯}_m\otimes {𝒯}_n$. Then $\varphi (A\otimes B)$ is an idempotent matrix in ${𝒯}_m\otimes {𝒯}_n$ whenever $A\otimes B$ is an idempotent matrix in ${𝒯}_m\otimes {𝒯}_n$ if and only if there exists an invertible matrix $P\in {𝒯}_m\otimes {𝒯}_n$ such that $\varphi (A\otimes B)=P({\xi }_{\mathrm{1}}(A)\otimes {\xi }_{\mathrm{2}}(B))P^{-\mathrm{1}},\forall A\in {𝒯}_m,B\in {𝒯}_n$, or when $m=n$, $\varphi (A\otimes B)=P({\xi }_{\mathrm{1}}(B)\otimes {\xi }_{\mathrm{2}}(A))P^{-\mathrm{1}},\forall A\in {𝒯}_m,B\in {𝒯}_m$, where ${\xi }_{\mathrm{1}}([a_{ij}])=[a_{ij}]$ or ${\xi }_{\mathrm{1}}([a_{ij}])=[a_{m-i+\mathrm{1},m-j+\mathrm{1}}]$ and ${\xi }_{\mathrm{2}}([b_{ij}])=[b_{ij}]$ or ${\xi }_{\mathrm{2}}([b_{ij}])=[b_{n-i+\mathrm{1},n-j+\mathrm{1}}].$

Let $ℳ$ and $𝒩$ be von Neumann algebras without central summands of type $I_1$. Assume that $\xi \in ℂ$ with $\xi \ne 1$. In this paper, all maps $\mathrm{\Phi }:ℳ\to 𝒩$ satisfying $\mathrm{\Phi }\left(AB-\xi BA\right)=\mathrm{\Phi }\left(A\right)\mathrm{\Phi }\left(B\right)-\xi \mathrm{\Phi }\left(B\right)\mathrm{\Phi }(A)$ are characterized.

We describe the structure of Lie triple derivations on $𝒥$-subspace lattice algebras. The results can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras, respectively.

We study maps $\varphi$ of positive operators of the Schatten $p$-classes (), which preserve the $p$-norms of convex combinations, that is, ${\parallel t\rho +(\mathrm{1}-t)\sigma \parallel }_p={\parallel t\varphi (\rho )+(\mathrm{1}-t)\varphi (\sigma )\parallel }_p,\forall \rho ,\sigma \in {𝒮}_p^{+}{(H)}_1,t\in [\mathrm{0,1}]$. They are exactly those carrying the form $\varphi (\rho )=U\rho U^{\mathrm{*}}$ for a unitary or antiunitary $U$. In the case $p=\mathrm{2}$, we have the same conclusion whenever it just holds ${\parallel \rho +\sigma \parallel }_2={\parallel \varphi (\rho )+\varphi (\sigma )\parallel }_2$ for all the positive Hilbert-Schmidt class operators $\rho ,\sigma$ of norm $\mathrm{1}$. Some examples are demonstrated.