In this article, the question of whether the Löwner partial order on the positive cone of an operator algebra is determined by the norm of any arbitrary Kubo–Ando mean is studied. The question was affirmatively answered for certain classes of Kubo–Ando means, yet the general case was left as an open problem. We here give a complete answer to this question, by showing that the norm of every symmetric Kubo–Ando mean is order-determining, i.e., if $A,B\in \mathcal B(H)^{++}$ satisfy $\Vert A\sigma X\Vert \le \Vert B\sigma X\Vert $ for every $X\in \mathcal {A}^{{++}}$, where $\mathcal A$ is the C*-subalgebra generated by $B-A$ and I, then $A\le B$.

We aim to characterise those transformations on the set of density operators (which are the mathematical representatives of the states in quantum information theory) that preserve a so-called generalised entropy of one fixed convex combination of operators. The characterisation strengthens a recent result of Karder and Petek where the preservation of the same quantity was assumed for all convex combinations.

For ${{C}^{*}}$-algebras $\mathcal{A}$ of real rank zero, we describe linear maps $\phi $ on $\mathcal{A}$ that are surjective up to ideals $\mathcal{J}$, and $\text{ }\pi \text{ (}A\text{)}$ is invertible in $\mathcal{A}/\mathcal{J}$ if and only if $\text{ }\pi \text{ (}\phi (A))$ is invertible in $\mathcal{A}/\mathcal{J}$, where $A\,\in \,\mathcal{A}$ and $\text{ }\pi \text{ }\text{:}\mathcal{A}\to \mathcal{A}\text{/}\mathcal{J}$ is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.