arXiv subjects

Lajos Molnár

Publications and source records attributed to Lajos Molnár.

10 recordsLinked to original sources

arXiv2026

On interrelations among different versions of a Heron type mean and commutativity in $C^*$-Algebras

The extension of the concept of a mean of positive real numbers to noncommutative settings, e.g., for Hilbert space operators, is a widely studied question. For example, in quantum information science, it is an important issue to find such extensions that fit the "best" to the studied physical problems. In fact, typically, there are many different ways of extension which, for commuting variables, all give the same value. In this paper, we are concerned with the converse: to what extent the coincidence of two extensions determines commutativity. Concretely, in our present work, we consider three different versions of the most common Heron type mean on the positive definite cone of a $C^*$-algebra: the Kubo-Ando type Heron mean, the naive or conventional version of the Heron mean, and the Wasserstein mean. We study equality relations among those objects and verify that they are closely connected to certain commutativity properties. They characterize either the commutativity of particular pairs of elements of a positive definite cone, or the centrality of positive definite elements, or the commutativity of the underlying algebra.

math.FA↗

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↗

arXiv2019

Maps on positive cones in operator algebras preserving power means

In this paper we consider power means of positive Hilbert space operators both in the conventional and in the Kubo-Ando senses. We describe the corresponding isomorphisms (bijective transformations respecting those means as binary operations) on positive definite cones and on positive semidefinite cones in operator algebras. We also investigate the question when those two sorts of power means can be transformed into each other.

math.FA↗

arXiv2019

Quantum Rényi relative entropies on density spaces of $C^*$-algebras: their symmetries and their essential difference

We extend the definitions of different types of quantum Rényi relative entropy from the finite dimensional setting of density matrices to density spaces of $C^*$-algebras. We show that those quantities (which trivially coincide in the classical commutative case) are essentially different on non-commutative algebras in the sense that none of them can be transformed to another one by any surjective transformation between density spaces. Besides, we determine the symmetry groups of density spaces corresponding to each of those quantum Rényi relative entropies and find that they are identical. Similar results concerning the Umegaki and the Belavkin-Staszewksi relative entropies are also presented.

math.OA↗

arXiv2019

On 2-local *-automorphisms and 2-local isometries of B(H)

It is an important result of \v Semrl which states that every 2-local automorphism of the full operator algebra over a separable Hilbert space is necessarily an automorphism. In this paper we strengthen that result quite substantially for *-automorphisms. Indeed, we show that one can compress the defining two equations of 2-local *-automorphisms into one single equation, hence weakening the requirement significantly, but still keeping essentially the conclusion that such maps are necessarily *-automorphisms.

math.FA↗

arXiv2017

Maps on positive definite operators preserving the quantum $χ_α^2$-divergence

We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum $χ_α^2$-divergence for some $α\in [0,1]$. We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.

math-ph↗

arXiv2015

Transformations on density operators and on positive definite operators preserving the quantum Rényi divergence

In a certain sense we generalize the recently introduced and extensively studied notion called quantum Rényi divergence (in another name, sandwiched Rényi relative entropy) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum Rényi divergence.

math.FA↗

arXiv2015

Maps on positive definite matrices preserving Bregman and Jensen divergences

In this paper we determine those bijective maps of the set of all positive definite $n\times n$ complex matrices which preserve a given Bregman divergence corresponding to a differentiable convex function that satisfies certain conditions. We cover the cases of the most important Bregman divergences and present the precise structure of the mentioned transformations. Similar results concerning Jensen divergences and their preservers are also given.

math.FA↗

arXiv2015

Continuous Jordan triple endomorphisms of $\mathbb{P}_2$

We describe the structure of all continuous Jordan triple endomorphisms of the set $\mathbb{P}_2$ of all positive definite $2\times 2$ matrices thus completing a recent result of ours. We also mention an application concerning sorts of surjective generalized isometries on $\mathbb{P}_2$ and, as second application, we complete another former result of ours on the structure of sequential endomorphisms of finite dimensional effect algebras.

math.FA↗

arXiv2015

On algebraic endomorphisms of the Einstein gyrogroup

We describe the structure of all continuous algebraic endomorphisms of the open unit ball $\mathbf{B}$ of $\mathbb{R}^3$ equipped with the Einstein velocity addition. We show that any nonzero such transformation originates from an orthogonal linear transformation on $\mathbb{R}^3$.

math-ph↗