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
We study the class of all groups in which the centralizer of each element is a subnormal subgroup. In particular, we focus on the case when the defect of every centralizer is at most 2. We show that a group without involutions satisfies this property if and only if it is 3-Engel.
COBISS.SI-ID: 16556889
We prove that a continuous map $\phi$ defined on the set of all $n \times n$ Hermitian matrices preserving order in both directions is up to a translation a congruence transformation or a congruence transformation composed with the transposition.
COBISS.SI-ID: 16921433