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In this paper we consider certain algebraic properties concerning variants of the Heron mean on positive definite cones of general C ⁎ -algebras. Those variants are the Kubo-Ando type Heron mean and the Wasserstein mean. The main part of the investigation concerns associativity properties. We present a number of results that show how far operations related to those two kinds of means are from being associative. Many of our results can also be viewed as characterizations of central positive definite elements or as characterizations of commutative C ⁎ -algebras. [ABSTRACT FROM AUTHOR]

In this paper we study preserver transformations between positive definite cones in operator algebras relating the spectral geometric mean of Fiedler and Pták. We also discuss kinds of structural similarities and dissimilarities between the Kubo-Ando geometric mean and the spectral geometric mean. [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 precise structure of all surjective ('a priori' nonlinear or, better say, nonaffine) Bures isometries between density spaces of C ⁎ -algebras equipped with faithful traces. It turns out that those maps are closely related to (linear) Jordan *-isomorphisms between the underlying algebras. Beside density spaces, we consider the problem also on the positive definite (and positive semidefinite) cones of C ⁎ -algebras. [ABSTRACT FROM AUTHOR]

In this paper we determine those bijective maps of the set of all positive definite n × 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. [ABSTRACT FROM AUTHOR]

We substantially extend and unify former results on the structure of surjective isometries of spaces of positive definite matrices obtained in the paper [14] . The isometries there correspond to certain geodesic distances in Finsler-type structures and to a recently defined interesting metric which also follows a non-Euclidean geometry. The novelty in our present paper is that here we consider not only true metrics but also so-called generalized distance measures which are parameterized by unitarily invariant norms and continuous real functions satisfying certain conditions. Among the many possible applications, we shall see that using our new result it is easy to describe the surjective maps of the set of positive definite matrices that preserve the Stein loss or several other types of divergences. We also present results concerning similar preserver transformations defined on the subset of all complex positive definite matrices with unit determinant. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present the general form of all continuous endomorphisms of the group of complex unitary matrices with respect to the Jordan triple product. These are the continuous maps which satisfy The result is applied to determine the structure of certain isometries of . These include the isometries relative to any metric given by a unitarily invariant norm on the space of all complex matrices and also the isometries relative to any member of a new class of metrics on recently introduced by Chau, Li, Poon and Sze [6]. [Copyright &y& Elsevier]

Abstract: We first determine the order automorphisms of the set of all positive definite operators with respect to the usual order and to the so-called chaotic order. We then apply those results to the following problems: (1) description of all bijective transformations on the space of nonsingular density operators (quantum states) which preserve the Umegaki or the Belavkin–Staszewski relative entropy; (2) characterization of the logarithmic product as the essentially unique binary operation on the set of positive definite operators that makes it an ordered commutative group with respect to the chaotic order. [Copyright &y& Elsevier]

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]

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]

Abstract: In this paper we consider multiplicative Jordan triple isomorphisms between the sets of self-adjoint elements (respectively the sets of positive elements) of von Neumann algebras. These transformations are the bijective maps which satisfy the equalityon their domains. We show that all those transformations originate from linear ∗-algebra isomorphisms and linear ∗-algebra antiisomorphisms in the case when the underlying von Neumann algebras do not have commutative direct summands. An application of our results concerning non-linear maps which preserve the absolute value of products is also presented. [Copyright &y& Elsevier]

Abstract: In this paper we consider two physically relevant numerical quantities (namely, the concurrence and Θ-fidelity) and an operation (called ♢-product) defined on the set of all quantum states or density operators. Our main aim is to present the complete descriptions of all bijective transformations which preserve any of these quantities or operation. [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]

In this note we describe the form of all those transformations (no linearity is assumed) of the set of all self-adjoint bounded linear operators acting on a Hilbert space which preserve commutativity in both directions and which are multiplicative on commuting pairs of operators. [Copyright &y& Elsevier]

In this paper we consider three operations on positive definite cones of C ⁎ -algebras which are related to weighted geometric means and appear in the formulas defining various versions of quantum Rényi relative entropy. We show how Jordan *-isomorphisms between C ⁎ -algebras can be characterized by the preservation of the norms of products under those operations or by the preservation of the operations themselves. We also obtain conditions for the commutativity of the underlying algebras by showing that we have that property if one of the quantities under considerations can be transformed by a surjective map to another different such quantity. [ABSTRACT FROM AUTHOR]