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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 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]

Abstract In this paper we present some characterizations for quasi-arithmetic operator means (among them the arithmetic and harmonic means) on the positive definite cone of the full algebra of Hilbert space operators, and also for the Kubo–Ando geometric mean on the positive definite cone of a general non-commutative C ⁎ -algebra. [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]

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

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 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]

Abstract: Let H be an infinite dimensional separable complex Hilbert space and be the group of all unitary operators on H. Motivated by the algebraic properties of surjective isometries of that have recently been revealed, and also by some classical results related to automorphisms of the unitary groups of operator algebras, we determine the structures of bijective transformations of that respect certain algebraic operations. These are, among others, the usual product of operators, the Jordan triple product, the inverted Jordan triple product, and the multiplicative commutator. Our basic approach to obtain these results is the use of commutativity preserving transformations on the unitary group. [Copyright &y& Elsevier]

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]

Abstract: Let be a finite-dimensional complex Hilbert space. The aim of this paper is to prove that every transformation on the space of all density operators on which preserves the relative entropy is implemented by either a unitary or an antiunitary operator on . [Copyright &y& Elsevier]

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]

Abstract: Let be a complex Hilbert space. The symbol stands for the harmonic mean of the positive bounded linear operators on in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to that operation. We prove that any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on . Similar results concerning the parallel sum and the arithmetic mean in the place of the harmonic mean are also presented. [Copyright &y& Elsevier]

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]

Abstract: We define a natural metric on the space of all bounded frame functions on a given Hilbert space and also on the space of all bounded frame functions with a given weight. We present the complete description of the surjective isometries of the so-obtained metric spaces. In fact, it turns out that those isometries are intrinsically induced by unitary or antiunitary operators on the underlying Hilbert space. [Copyright &y& Elsevier]

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]

Abstract: Let H be a Hilbert space and let and be standard ∗-operator algebras on H. Denote by and the set of all self-adjoint operators in and , respectively. Assume that and are surjective maps such that and for every pair , . Then there exist an invertible bounded linear or conjugate-linear operator and a constant such that , , and , . [Copyright &y& Elsevier]

Abstract: In this paper we introduce some natural numerical quantities which measure differentkinds of correspondences between Hilbert space effects. We show that every objective map of the space ɛ(H) of all effects on the Hilbert space H which preserves any of these quantities is implemented by a unitary or antiunitary operator on H. [Copyright &y& Elsevier]

In this paper we study a new class of transformations on the set of all Hilbert space effects. This consists of the bijective maps which preserve the order and zero product in both directions. The main result of the paper gives a complete description of the structure of those transformations. As applications we obtain additional new results and some former ones as easy corollaries. In particular, we obtain the form of the ortho-order automorphisms as well as that of the sequential automorphisms. In the last paragraph of the paper we show that the automorphisms of these two kinds belong to our class of transformations even when their domain is the set of all effects in a general von Neumann algebra. [Copyright &y& Elsevier]

It is well known that the ⊥-order automorphisms of the effect algebra of a Hilbert space of dimension not less than three are implemented by unitary or antiunitary operators. The aim of this paper is to show that the same assertion also holds true in the two-dimensional case. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Abstract: Let be a complex Hilbert space, n a given positive integer and let be the set of all projections on with rank n. Under the condition , we describe the surjective isometries of with respect to the gap metric (the metric induced by the operator norm). [Copyright &y& Elsevier]