Let $A$ and $B$ be unital semisimple Banach algebras. If $\phi \colon M_2(A)\to B$ is a bijective spectrum-preserving linear map, then $\phi$ is a Jordan homomorphism.
COBISS.SI-ID: 16067673
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 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
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
The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to $\mathbb{C}$. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes it possible for us to extend Frobenius' and Zorn's theorems in a way that it also involves the fifth Cayley-Dickson algebra, the sedenions.
COBISS.SI-ID: 15758681