Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all $m\times n$ matrices over a division ring $\mathbb{D}$ which preserve adjacency in both directions. Motivated by several applications we study a long standing open problem of possible improvements. There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes? A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings we solve all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings we get such an optimal result only for square matrices and give examples showing that it cannot be extended to the non-square case.
COBISS.SI-ID: 16947545
We describe the general form of bijective comparability preserving transformations of the Hilbert space effect algebra, thus improving several known characterizations of ortho-order automorphisms.
COBISS.SI-ID: 16568409
Let $H$ be a Hilbert space and $E(H)$ the effect algebra on $H$, that is, $E(H)$ is the set of all self-adjoint operators $A \colon H \to H$ satisfying $0 \leqslant A \leqslant I$. The effect algebra can be equipped with several operations and relations that are important in mathematical foundations of quantum mechanics. Automorphisms with respect to these operations or relations are called symmetries. We present a new method that can be used to describe the general form of such maps. The main idea is to reduce this kind of problem to the study of adjacency-preserving maps. The efficiency of this approach is illustrated by reproving some known results as well as by obtaining some new theorems.
COBISS.SI-ID: 16756569
Self-adjoint operators represent bounded observables in mathematical foundations of quantum mechanics. The set of all self-adjoint operators can be equipped with several operations and relations having important interpretations in physics. Automorphisms with respect to these relations or operations are called symmetries. Many of them turn out to be real-linear up to a translation. We present a unified approach to the description of the general form of such symmetries based on adjacency preserving maps. We consider also symmetries defined on the set of all positive operators or on the set of all positive invertible operators. In particular, we will see that the structural result for adjacency preserving maps on the set of all positive invertible operators differs a lot from its counterpart on the set of all selfadjoint operators.
COBISS.SI-ID: 16568665
We solve Kaplansky's problem concerning the structure of linear preservers of invertibility in the special case of maps on central simple algebras.
COBISS.SI-ID: 16962649