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We extend the definitions of different types of quantum Rényi relative entropy from the finite dimensional setting of density matrices to density spaces of C ⁎ -algebras. We show that those quantities (which trivially coincide in the classical commutative case) are essentially different on non-commutative algebras in the sense that none of them can be transformed to another one by any surjective transformation between density spaces. Besides, we determine the symmetry groups of density spaces corresponding to each of those quantum Rényi relative entropies and find that they are identical. Similar results concerning the Umegaki and the Belavkin-Staszewksi relative entropies are also presented. [ABSTRACT FROM AUTHOR]

It is an important result of Šemrl which states that every 2-local automorphism of the full operator algebra over a separable Hilbert space is necessarily an automorphism. In this paper we strengthen that result quite substantially for *-automorphisms. Indeed, we show that one can compress the defining two equations of 2-local *-automorphisms into one single equation, hence weakening the requirement significantly, but still keeping essentially the conclusion that such maps are necessarily *-automorphisms. [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 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]

We investigate the algebraic properties of the operation ab=aba on the set of all positive invertible elements of a C\*-algebra A. We show that its commutativity, associativity and distributivity are each equivalent to the commutativity of A. We present abstract characterizations of the operation and a few related ones, too. [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]

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]

Abstract: In this paper we present the complete description of surjective isometries of the space of all probability distribution functions on with respect to the Lévy metric. [Copyright &y& Elsevier]

Abstract: In this paper we describe the general forms of surjective isometries of the space of all probability distribution functions on with respect to the Kolmogorov–Smirnov metric. [Copyright &y& Elsevier]

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]

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]

In this paper we determine all the bijective linear maps on the space of bounded observables which preserve a fixed moment or the variance. Nonlinear versions of the corresponding results are also presented. [Copyright &y& Elsevier]

We present an analogue of Uhlhorn''s version of Wigner''s theorem on symmetry transformations for the case of indefinite inner product spaces. This significantly generalizes a result of Van den Broek. The proof is based on our main theorem, which describes the form of all bijective transformations on the set of all rank-one idempotents of a Banach space which preserve zero products in both directions. [Copyright &y& Elsevier]

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]

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]