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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 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]

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 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]

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 those bijective maps on the cone of all positive invertible elements of a C ⁎ -algebra with a normalized faithful trace which preserve certain kinds of quasi-entropy. It is shown that essentially any such map is equal to a Jordan *-isomorphism of the underlying algebra multiplied by a central positive invertible element. [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]

Abstract: We show that the existence of a surjective isometry (which is merely a distance preserving map) between the unitary groups of unital -algebras implies the existence of a Jordan *-isomorphism between the algebras. In the case of von Neumann algebras we describe the structure of those isometries showing that any of them is extendible to a real linear Jordan *-isomorphism between the underlying algebras multiplied by a fixed unitary element. We present a result of similar spirit for the surjective Thompson isometries between the spaces of all invertible positive elements in unital -algebras. [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: 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]

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]

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]