Search

This year the journal Acta Scientiarum Mathematicarum, which is commonly called Acta Szeged, celebrates its 100th anniversary. In this article, we provide a brief overview of the past and the present of the journal, which is currently one of the 20 oldest and still active mathematical periodicals worldwide. [ABSTRACT FROM AUTHOR]

In this paper we consider some important non Kubo–Ando means on positive definite cones of C ∗ -algebras and investigate their relations to the usual (Löwner) order. We study two basic questions. First, when, on the positive definite cone of a C ∗ -algebra, does such a mean σ satisfy the inequality A ≤ A σ B ≤ B for any pair A, B of elements with A ≤ B . This requirement seems to be the most basic one a mean should fulfill even for numbers. Actually, we study stronger forms of that question and obtain results characterizing central elements and commutativity of C ∗ -algebras. Second, we investigate monotonicity properties of those means with respect to the usual order which also turn to be closely related to the commutativity of the underlying algebras. [ABSTRACT FROM AUTHOR]

A foreword to "Acta Scientiarum Mathematicarum" is presented.

The primary aim of this paper is to present the complete description of the isomorphisms between positive definite cones of C^*-algebras with respect to the recently introduced Wasserstein mean and to show the nonexistence of nonconstant such morphisms into the positive reals in the case of von Neumann algebras without type I_2, I_1 direct summands. A comment on the algebraic properties of the Wasserstein mean relating associativity is also made. [ABSTRACT FROM AUTHOR]

In a very general setting, we introduce a new type of local maps, a new sort of reflexive closure of a given class of transformations relative to a given operation that we call operational reflexive closure, and a corresponding concept of reflexivity. We calculate the operational reflexive closures of some important classes of transformations and significantly strengthen former 2-reflexivity results concerning the automorphism groups of various operator structures. A typical new result is this: if ø is a map from the unitary group over a separable infinite dimensional Hilbert space into itself with the property that for any pair V, W of unitaries there is a group automorphism αV,W of the unitary group such that ø(V0 ø(W) = αV,W (VW), then either ø itself or ø is a group automorphism. This result substantially generalizes a former one on the 2-reflexivity of the automorphism group of the unitary group. We also present open problems and questions for further study. [ABSTRACT FROM AUTHOR]

In this paper we deal with the question of when the norm of a Kubo–Ando mean σ determines the order on the positive definite cone of an operator algebra. More precisely, we investigate the following problem: when, for any given pair A, B of positive definite elements of the algebra, we have that A ≤ B holds if and only if ‖ A σ X ‖ ≤ ‖ B σ X ‖ is valid for all positive definite elements X. We present several results and formulate related open problems. [ABSTRACT FROM AUTHOR]

In this paper we consider some faithful representations of positive Hilbert space operators on structures of nonnegative real functions defined on the unit sphere of the Hilbert space in question. Those representations turn order relations between positive operators to order relations between real functions. Two of them turn the usual Löwner order between operators to the pointwise order between functions, another two turn the spectral order between operators to the same, pointwise order between functions. We investigate which algebraic operations those representations preserve, hence which kind of algebraic structure the representing functions have. We study the differences among the different representing functions of the same positive operator. Finally, we introduce a new complete metric (which corresponds naturally to two of those representations) on the set of all invertible positive operators and formulate a conjecture concerning the corresponding isometry group. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove that any bijective map between the positive definite cones of von Neumann algebras which preserves a certain unitarily invariant norm of a particular weighted geometric mean of elements is essentially (up to two-sided multiplication by an invertible positive element) equal to the restriction of a Jordan *-isomorphism between the algebras. [ABSTRACT FROM AUTHOR]

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 $$\chi _\alpha ^2$$ -divergence for some $$\alpha \in [0,1]$$ . 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. [ABSTRACT FROM AUTHOR]

Wu [‘An order characterization of commutativity for $C^{\ast }$-algebras’, Proc. Amer. Math. Soc.129 (2001), 983–987] proved that if the exponential function on the set of all positive elements of a $C^{\ast }$-algebra is monotone in the usual partial order, then the algebra in question is necessarily commutative. In this note, we present a local version of that result and obtain a characterisation of central elements in $C^{\ast }$-algebras in terms of the order. [ABSTRACT FROM PUBLISHER]

In this short note we present a common characterisation of the logarithmic function and the subspace of all trace zero elements in finite von Neumann factors. [ABSTRACT FROM AUTHOR]

In the earlier paper (Molnár and Šemrl in Lett Math Phys 80:239-255, ), we described the structure of all spectral order automorphisms of the sets of Hilbert space effects and bounded observables in the case where the dimension of the underlying Hilbert space is at least 3. The aim of this note is to present a complete description in the missing two-dimensional case. We will see that in that case there is a one-to-one correspondence between the set of all spectral order automorphisms and the set of all bijective maps of pure states together with the set of all strictly increasing bijections of the real unit interval or the real line. [ABSTRACT FROM AUTHOR]

In this paper we determine the structure of all so-called generalized isometries of the special unitary group which are transformations that respect any member of a large collection of generalized distance measures. [ABSTRACT FROM AUTHOR]

We prove that a bijective linear transformation between von Neumann factors that commute with a λ-Aluthge transform is necessarily a non-zero scalar multiple of an algebra *-isomorphism in the case of algebras that are not of type I2. As for type I2 factors, that is, in the particular case of the algebra of 2 by 2 complex matrices, we also present a complete description of those transformations which is a bit different. Namely, non-zero scalar multiples of the sum of an algebra *-antiisomorphism and the negative of the trace functional times the identity also show up. [ABSTRACT FROM AUTHOR]

Motivated by recent investigations on norm-additive and spectrally multiplicative maps on various spaces of functions, in this paper we determine all bijective transformations between the positive cones of standard operator algebras over a Hilbert space which preserve a given symmetric norm of a given mean of elements. A result of similar spirit is also presented concerning transformations between cones of nonnegative elements of certain algebras of continuous functions. [ABSTRACT FROM AUTHOR]

We describe the structure of all continuous algebraic endomorphisms of the open unit ball B of R³ equipped with the Einstein velocity addition. We show that any nonzero such transformation originates from an orthogonal linear transformation on R³. [ABSTRACT FROM AUTHOR]

We prove that there is no bijective map between the set of all positive definite operators and the set of all self-adjoint operators on a Hilbert space with dimension greater than 1 which preserves the usual order (the one coming from the concept of positive semidefiniteness) in both directions. We conjecture that a similar assertion is true for general noncommutative C*-algebras and present a proof in the finite dimensional case. [ABSTRACT FROM AUTHOR]

In this paper, we determine the structure of certain algebraic morphisms and isometries of the spaceof allcomplex positive definite matrices. In the case, we describe all continuous Jordan triple endomorphisms ofwhich are continuous mapssatisfyingIt has recently been discovered that surjective isometries of certain substructures of groups equipped with metrics which are in a way compatible with the group operations have algebraic properties that relate them rather closely to Jordan triple morphisms. This makes us possible to use our structural results to describe all surjective isometries ofthat correspond to any member of a large class of metrics generalizing the geodesic distance in the natural Riemannian structure on. Finally, we determine the isometry group ofrelative to a very recently introduced metric that originates from the divergence called Stein’s loss. [ABSTRACT FROM AUTHOR]

For a given probability distribution λ,..., λ we determine the structure of all such maps defined on a dense subset of density operators which leave the Holevo bound invariant i.e. which satisfy for all possible collections ρ,..., ρ of density operators. [ABSTRACT FROM AUTHOR]

In this note we show that the bilocal *-automorphisms of the C*-algebra B( H) of all bounded linear operators acting on a complex infinite dimensional separable Hilbert space H are precisely the unital algebra *-endomorphisms of B( H). [ABSTRACT FROM AUTHOR]

We present a few characterizations of the commutativity of C*-algebras in terms of particular algebraic properties of power functions, the logarithmic and exponential functions, and the sine and cosine functions. [ABSTRACT FROM AUTHOR]

For an arbitrary strictly convex function $$f$$ defined on the non-negative real line we determine the structure of all transformations on the set of density operators which preserve the quantum $$f$$-divergence. [ABSTRACT FROM AUTHOR]

In this article we consider the set of all positive semidefinite linear operators on a finite-dimensional complex Hilbert space equipped with the so-called logarithmic product. We describe the general form of all automorphisms of this structure which are continuous at 0. [ABSTRACT FROM AUTHOR]

In this paper we describe all surjective isometries of the unitary group of a complex Hilbert space. A result on Thompson isometries of the space of all invertible positive elements of a unital C*-algebra is also presented. [ABSTRACT FROM AUTHOR]

In this article we consider certain kinds of metrics and relative entropies on sets of density operators acting on a finite-dimensional complex Hilbert space. The metrics are the ones induced by the von Neumann–Schatten p-norms. We describe the general form of ‘a priori’ nonsurjective isometries of the space of all density operators with respect to those distances. In the second part of this article we study mappings preserving entropic quantities. We determine the structure of ‘a priori’ nonsurjective maps on the set of all density operators which leave a certain measure of relative entropy invariant and also characterize the surjective maps on the set of all invertible density operators with similar invariance properties. [ABSTRACT FROM PUBLISHER]

In this paper we describe the structure of all bijective nonlinear maps on the space of all bounded self-adjoint operators acting on a complex separable Hilbert space of dimension at least 3 which preserve a measure of commutativity, namely, the norm of the commutator of operators. [ABSTRACT FROM AUTHOR]

In this paper we describe the structure of surjective isometries of the spaces of all absolutely continuous, singular, or discrete probability distribution functions on R equipped with the Kolmogorov-Smirnov metric. We also study the structure of affine automorphisms of the space of all distribution functions. [ABSTRACT FROM AUTHOR]

In this paper we describe the structure of all bijective linear maps between the spaces of self-adjoint elements of von Neumann algebras that preserve the so-called maximal deviation. It turns out that those transformations are closely related to Jordan *-isomorphisms. [ABSTRACT FROM PUBLISHER]

Let H be a finite dimensional Hilbert space. The aim of this short note is to prove that every bijective map on the space S(H) of all density operators on H which preserves the relative entropy is of the form [lowercase_phi_synonym](ρ)=UρU* with some unitary or antiunitary operator U on H. [ABSTRACT FROM AUTHOR]

We describe the general form of bijective maps on the space of all Hilbert space effects or all observables on a finite dimensional space which preserve the spectral order in both directions. [ABSTRACT FROM AUTHOR]

In this article after defining the abstract concept of compatibility-like functions on quantum states, we prove that every bijective transformation on the set of all states which preserves such a function is implemented by an either unitary or antiunitary operator. [ABSTRACT FROM AUTHOR]

We present a new simple proof of the structural result for elementary operators on standard operator algebras. Using the idea of this short proof we can characterize surjective maps between standard operator algebras having a certain multiplicativity-like property appearing in the abstract definition of elementary operators of length one. In particular, we show that such maps are automatically additive. [ABSTRACT FROM AUTHOR]

Let H be a separable infinite-dimensional complex Hilbert space. We prove that every continuous 2-local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented. [ABSTRACT FROM AUTHOR]

In this paper we characterize the automorphisms of Hilbert space effect algebras by means of their preserving properties which concern certain relations and quantities appearing in quantum measurement theory. [ABSTRACT FROM AUTHOR]

Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the following way: Every bijective transformation on the set ℒ of all 1-dimensional subspaces of a Hilbert space H which preserves the angle between the elements of ℒ is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n∈ℕ fixed. [ABSTRACT FROM AUTHOR]

Wigner's theorem on symmetry transformations can be formulated in the followingway. If φ is a bijective map on the set of all nonzero minimal projections in atype I factor A which preserves transition probabilities with respect to a faithfulnormal semifinite trace, then it can be extended to a linear *-automorphism orto a linear *-antiautomorphism of A. In this paper we prove a natural analogueof this statement for type II factors. [ABSTRACT FROM AUTHOR]

We present a generalization of Wigner's unitary-antiunitary theorem for pairs of ray transformations. As a particular case, we get a new Wigner-type theorem for non-Hermitian indefinite inner product spaces. [ABSTRACT FROM AUTHOR]

The set of all effects on a Hilbert space has an affine structure (it is a convex set) as well as a multiplicative structure (it can be equipped with the so-called Jordan triple product). In this paper we describe the corresponding automorphisms of that set. [ABSTRACT FROM AUTHOR]

The aim of this paper is to solve a linear preserver problem on the function algebra C( X). We show that in the case in which X is a first countable compact Hausdorff space, every linear bijection $\phi :C(X)\to C(X)$ having the property that $\hbox {diam} (\phi (f)(X))=\hbox {diam} (f(X)) (f\in C(X))$ is of the form¶¶ $ \phi (f)=\tau \cdot f\circ \varphi +t(f)1 \,\, (f\in C(X))$ ¶¶where $\tau \in {\Bbb C}, |\tau |=1, $\varphi :X\to X$ is a homeomorphism and $t:C(X)\to {\Bbb C}$ is a linear functional. [ABSTRACT FROM AUTHOR]

Let H be a separable infinite-dimensional complex Hilbert space. We prove that if Ф: ℬ( H)→ℬ( H) is a-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then Ф is an isometric linear or conjugate-linear algebra automorphism of ℬ( H). In particular, if the unilateral shift is contained in the range of a-endomorphism Ф of ℬ( H), then Ф is bijective. [ABSTRACT FROM AUTHOR]