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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.

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]

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]

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]

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]

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 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]

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]

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]

In this paper we give the general form of order isomorphisms and that of triple isomorphisms of standard operator algebras. For any Schatten- von Neumann class of compact operators as well as for the whole operator algebra, it is proved that the group of all of its order automorphisms is reflexive in the algebra of all linear (not necessarily continuous) selfmaps of the underlying operator algebra. A similar reflexivity result concerning triple automorphisms of an arbitrary operator ideal is also presented. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the authors discuss various reports within the issue including topics on mathematical structure, models, and mechanics.