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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 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]

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 a bijective linear transformation between von Neumann factors that commute with a λ-Aluthge transform is necessarily a non-zero scalar multiple of an algebra *-isomorphism in the case of algebras that are not of type I2. As for type I2 factors, that is, in the particular case of the algebra of 2 by 2 complex matrices, we also present a complete description of those transformations which is a bit different. Namely, non-zero scalar multiples of the sum of an algebra *-antiisomorphism and the negative of the trace functional times the identity also show up. [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]

We present a few characterizations of the commutativity of C*-algebras in terms of particular algebraic properties of power functions, the logarithmic and exponential functions, and the sine and cosine functions. [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: 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: 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: 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 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]

An introduction is presented in which the authors discuss various reports within the issue including topics on mathematical structure, models, and mechanics.