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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]

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]

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]

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]

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]

Abstract: We establish the algebraic reflexivity of three isometry groups of operator structures: the group of all surjective isometries on the unitary group, the group of all surjective isometries on the set of all positive invertible operators equipped with the Thompson metric, and the group of all surjective isometries on the general linear group of , the operator algebra over a complex infinite dimensional separable Hilbert space . We show that those isometry groups coincide with certain groups of automorphisms of corresponding structures and hence we also obtain the reflexivity of some automorphism groups. [Copyright &y& Elsevier]