arXiv subjects

Peter Šemrl

Publications and source records attributed to Peter Šemrl.

10 recordsLinked to original sources

arXiv2026

Local automorphisms of some classical groups

A map on a group into itself is called a local automorphism if at any two points of the group, it can be interpolated by an automorphism of that group. In this paper we investigate the question of how local automorphisms of some classical groups are related to automorphisms. In some cases it turns out that the local automorphisms are in fact automorphisms. In the remaining cases we show that the local automorphisms are still closely related to the automorphisms.

math.GR↗

arXiv2025

Optimal version of the fundamental theorem of chronogeometry

We study lightlikeness preserving mappings from the $4$-dimensional Minkowski spacetime $\mathcal{M}_4$ to itself under no additional regularity assumptions like continuity, surjectivity, or injectivity. We prove that such a mapping $ϕ$ satisfies one of the following three conditions. (1) The mapping $ϕ$ can be written as a composition of a Lorentz transformation, a multiplication by a positive scalar, and a translation. (2) There is an event $r\in \mathcal{M}_4$ such that $ϕ(\mathcal{M}_4\setminus\{r\})$ is contained in one light cone. (3) There is a lightlike line $\ell$ such that $ϕ(\mathcal{M}_4\setminus \ell)$ is contained in another lightlike line. Here, a line that is contained in some light cone in $\mathcal{M}_4$ is called a lightlike line. We also give several similar results on mappings defined on a certain subset of $\mathcal{M}_4$ or the compactification of $\mathcal{M}_4$.

math-ph↗

arXiv2023

Nonexpansive and noncontractive mappings on the set of quantum pure states

Wigner's theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry nonexpansive maps and noncontractive maps are well studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such characterizations are injectivity or surjectivity and they differ in the finite and the infinite-dimensional case. Motivated by a recently obtained optimal version of Uhlhorn's generalization of Wigner's theorem, we also give a description of nonexpansive maps which satisfy a condition that is much weaker than surjectivity. Such maps do not need to be Wigner symmetries. The optimality of all presented results is shown by counterexamples.

math-ph↗

arXiv2023

The Waring Problem for Matrix Algebras, II

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and $α_1,α_2,α_3$ are nonzero elements from $F$ such that $α_1+α_2+α_3=0$, then every trace zero $n\times n$ matrix over $F$ can be written as $α_1 A_1+α_2A_2+α_3A_3$ for some $A_i$ in the image of $f$ in $M_n(F)$.

math.RA↗

arXiv2020

Loewner's theorem for maps on operator domains

The classical Loewner's theorem states that operator monotone functions on real intervals are described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains by biholomorphic automorphisms of the generalized upper half-plane, which is the collection of all operators with positive invertible imaginary part. We describe such maps in an explicit manner, and examine properties of maximal local order isomorphisms. Moreover, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.

math.FA↗

arXiv2019

Continuous coexistency preservers on effect algebras

Let $H$ be a finite-dimensional Hilbert space, $\dim H \ge 2$. We prove that every continuous coexistency preserving map on the effect algebra $E(H)$ is either a standard automorphism of $E(H)$, or a standard automorphism of $E(H)$ composed with the orthocomplementation. We present examples showing the optimality of the result.

math-ph↗

arXiv2016

Isometries of Grassmann spaces

Botelho, Jamison, and Moln\' ar have recently described the general form of surjective isometries of Grassmann spaces on complex Hilbert spaces under certain dimensionality assumptions. In this paper we provide a new approach to this problem which enables us first, to give a shorter proof and second, to remove dimensionality constraints completely. In one of the low dimensional cases, which was not covered by Botelho, Jamison, and Moln\' ar, an exceptional possibility occurs. As a byproduct, we are able to handle the real case as well. Furthermore, in finite dimensions we remove the surectivity assumption. A variety of tools is used in order to achieve our goal, such as topological, geometrical and linear algebra techniques. The famous two projections theorem for two finite rank projections will be re-proven using linear algebraic methods. A theorem of Györy and the second author on orthogonality preservers on Grassmann spaces will be strengthened as well. This latter result will be obtained by using Chow's fundamental theorem of geometry of Grassmannians.

math.FA↗

arXiv2015

Hua's fundamental theorem of geometry of rectangular matrices over EAS division rings

The fundamental theorem of geometry of rectangular 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. This result proved by Hua in the nineteen forties has been recently improved in several directions. One can study such maps without the bijectivity assumption or one can try to get the same conclusion under the weaker assumption that adjacency is preserved in one direction only. And the last possibility is to study maps acting between matrix spaces of different sizes. The optimal result would describe maps preserving adjacency in one direction only acting between spaces of rectangular matrices of different sizes in the absence of any regularity condition (injectivity or surjectivity). A division ring is said to be EAS if it is not isomorphic to any proper subring. It has been known before that it is possible to construct adjacency preserving maps with wild behaviour on matrices over division rings that are not EAS. For matrices over EAS division rings it has been recently proved that adjacency preserving maps acting between matrix spaces of different sizes satisfying a certain weak surjectivity condition are either degenerate or of the expected simple standard form. We will remove this weak surjectivity assumption, thus solving completely the long standing open problem of the optimal version of Hua's theorem.

math.RA↗

arXiv2015

Continuous space-time transformations

We prove that every continuous map acting on the four-dimensional Minkowski space and preserving light cones in one direction only is either a Poincaré similarity, that is, a product of a Lorentz transformation and a dilation, or it is of a very special degenerate form. In the presence of the continuity assumption the main tool in the proof is a basic result from the homotopy theory of spheres.

math.RA↗

arXiv2013

From geometry to invertibility preservers

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.

math.RA↗