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In this paper, we consider a (nonlinear) transformation $\Phi$ of invertible positive elements in $C^*$-algebras which preserves the norm of any of the three fundamental means of positive elements; namely, $\|\Phi(A)\mm \Phi(B)\| = \|A\mm B\|$, where $\mm$ stands for the arithmetic mean $A\nabla B=(A+B)/2$, the geometric mean $A\#B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$, or the harmonic mean $A!B=2(A^{-1} + B^{-1})^{-1}$. Assuming that $\Phi$ is surjective and preserves either the norm of the arithmetic mean or the norm of the geometric mean, we show that $\Phi$ extends to a Jordan $*$-isomorphism between the underlying full algebras. If $\Phi$ is surjective and preserves the norm of the harmonic mean, then we obtain the same conclusion in the special cases where the underlying algebras are $AW^*$-algebras or commutative $C^*$-algebras. In the commutative case, for a transformation $T: F(\mathrm{X})\subset C_0(\mathrm{X})_+\rightarrow C_0(\mathrm{Y})_+$, we can relax the surjectivity assumption and show that $T$ is a generalized composition operator if $T$ preserves the norm of the (arithmetic, geometric, harmonic, or in general any power) mean of any finite collection of positive functions, provided that the domain $F(\mathrm{X})$ contains sufficiently many elements to peak on compact $G_\delta$ sets. When the image $T(F(\mathrm{X}))$ also contains sufficiently many elements to peak on compact $G_\delta$ sets, $T$ extends to an algebra $*$-isomorphism between the underlying full function algebras., Comment: to appear in Joournal of Operator Theory

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., Comment: To appear in Aequationes Math

We extend the definitions of different types of quantum R\'enyi 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 R\'enyi relative entropies and find that they are identical. Similar results concerning the Umegaki and the Belavkin-Staszewksi relative entropies are also presented., Comment: To appear in J. Funct. Anal

It is an important result of \v Semrl 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., Comment: To appear in J. Math. Anal. Appl

In this paper we describe the structure of all sequential isomorphisms between the sets of von Neumann algebra effects. It turns out that if the underlying algebras have no commutative direct summands, then every sequential isomorphism between the sets of their effects extends to the direct sum of a *-isomorphism and a *-antiisomorphism between the underlying von Neumann algebras., Comment: To appear in Acta Sci. Math. (Szeged)

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 these two kinds of automorphisms belong to our class of transformations even when their domain is the set of all effects in a general von Neumann algebra., Comment: 22 pages, submitted for publication

In this paper after defining the abstract concept of compatibility-like functions on quantum states, we prove that every bijective transformation on the set of all states which preserves such a function is implemented by an either unitary or antiunitary operator., Comment: 11 pages, submitted for publication

This paper treats the isometries of metric spaces of quantum states. We consider two metrics on the set all quantum states, namely the Bures metric and the one which comes from the trace-norm. We describe all the corresponding (nonlinear) isometries and also present similar results concerning the space of all (non-normalized) density operators., Comment: 9 pages. To appear in J. Phys. A: Math. Gen

The aim of this paper is to show that if an order preserving bijective transformation of the Hilbert space effect algebra also preserves the probability with respect to a fixed pair of mixed states, then it is an ortho-order automorphism. A similar result for the orthomodular lattice of all sharp effects (i.e., projections) is also presented., Comment: 12 pages, to appear in Rep. Math. Phys

In this paper we extend a result of Semrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras of which statement we present a short proof. The need to get such a proof was formulated in Semrl's paper., Comment: 9 pages. To appear in Proc. Amer. Math. Soc

We prove that any bijective fidelity preserving transformation on the set of all density operators on a Hilbert space is implemented by an either unitary or antiunitary operator on the underlying Hilbert space., Comment: This is corrected version of the paper math.OA/0108060. The paper has already appeared in ROMP (vol. 48 (2001), 299-303)

Let H be a separable infinite dimensional complex Hilbert space. We prove that every continuous 2-local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented., Comment: 10 pages. To appear in Lett. Math. Phys

In this paper we characterize the automorphisms of Hilbert space effect algebras by means of their preserving properties which concern certain relations and quantities appearing in quantum measurement theory., Comment: 15 pages. To appear in Commun. Math. Phys

Jordan isomorphisms of rings are defined by two equations. The first one is the equation of additivity while the second one concerns multiplicativity with respect to the so-called Jordan product. In this paper we present results showing that on standard operator algebras over spaces with dimension at least 2, the bijective solutions of that second equation are automatically additive., Comment: 14 pages. To appear in Proceedings of the 38th ISFE, Kluwer Academic Publishers

Let H be a complex Hilbert space and denote by Bs(H) the set of all self-adjoint bounded linear operators on H. In this paper we describe the form of all bijective maps (no linearity or continuity is assumed) on Bs(H) which preserve the usual order in both directions., Comment: 19 pages. To appear in J. Math. Phys

We present some nonlinear characterizations of the automorphisms of the operator algebra $B(H)$ and the function algebra $C(X)$ by means of their spectrum preserving properties., Comment: 11 pages. To appear in Proc. Amer. Math. Soc

Under the mild condition of continuity at a single point we describe all the bijections of the set of all partial isometries on a Hilbert space which preserve the order and the orthogonality in both directions. Moreover, we present a natural analogue of Wigner's theorem on quantum mechanical symmetries for the set of all rank-1 partial isometries., Comment: 9 pages. To appear in Arch. Math

Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the following way: Every bijective transformation on the set L of all 1-dimensional subspaces of a Hilbert space H which preserves the angle between the elements of L is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n fixed., Comment: 20 pages. To appear in Commun. Math. Phys

We prove that every 2-local automorphism of the unitary group or the general linear group on a complex infinite-dimensional separable Hilbert space is an automorphism. Thus these types of transformations are completely determined by their local actions on the two-points subsets of the groups in question., Comment: 9 pages. To appear in Expo. Math

The set of all efects on a Hilbert space has an affine structure (it is a convex set) as well as a multiplicative structure (it can be equipped with the so-called Jordan triple product). In this paper we describe the corresponding automorphisms of that set., Comment: 9 pages. To appear in Lett. Math. Phys

The aim of this paper is to show that between standard operator algebras every bijective map with a certain multiplicativity property related to Jordan triple isomorphisms of associative rings is automatically additive., Comment: 8 pages. To appear in Studia Math

Let H be a complex infinite dimensional Hilbert space. We describe the form of all *-semigroup endomorphisms $\phi$ of B(H) which are uniformly continuous on every commutative C*-subalgebra. In particular, we obtain that if $\phi$ satisfies $\phi(0)=0$, then $\phi$ is additive., Comment: 7 pages. To appear in the Proceedings of the Memorial Conference for Bela Szokefalvi-Nagy, 1999, Szeged, Hungary

We present a generalization of Wigner's unitary-antiunitary theorem for pairs of ray transformations. As a particular case, we get a new Wigner-type theorem for non-Hermitian indefinite inner product spaces., Comment: 7 pages. To appear in Commun. Math. Phys

Let X be a compact Hausdorff space and let H be a separable Hilbert space. We prove that the group of all order automorphisms of the $C^*$-algebra $C(X)\otimes B(H)$ is algebraically reflexive., Comment: 8 pages. To appear in Proc. Amer. Math. Soc

Let H be a Hilbert $C^*$-module over a matrix algebra A. It is proved that any function $T:H\to H$ which preserves the absolute value of the (generalized) inner product is of the form $Tf=\phi(f)Uf$ $(f\in H)$, where $\phi$ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner's classical unitary-antiunitary theorem for Hilbert modules., Comment: 27 pages. To appear in J. Math. Phys

We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra B(H), H being a separable Hilbert space. Let $\phi:B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of B(H) (these automorphisms may, of course, depend on the point) in the operator norm. Then $\phi$ is an automorphism of the algebra B(H)., Comment: 8 pages. To appear in Acta Sci. Math. (Szeged)

In this paper we describe the form of those continuous multiplicative maps on B(H) (H being a separable complex Hilbert space of dimension not less than 3) which preserve the rank, or the corank. Furthermore, we characterize those continuous *-semigroup endomorphisms of B(H) which are spectrum non-increasing., Comment: 13 pages. To appear in Linear Algebra Appl

As a continuation of the work on linear maps between operator algebras which preserve certain subsets of operators with finite rank, or corank, here we consider the problem inbetween, that is, we treat the question of preserving operators with infinite rank and infinite corank. Since, as it turns out, in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems, hence we restrict our attention to certain important classes of operators like idempotents, or projections, or partial isometries. We conclude the paper with a result on the form of linear maps which preserve the left ideals in B(H)., Comment: 11 pages. To appear in Linear Algebra Appl

We present an operator algebraic approach to Wigner's unitary-antiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert modules over matrix algebras. We also present a Wigner-type result for maps on prime C*-algebras., Comment: AMSLaTeX, 17 pages. To appear in J. Austral. Math. Soc

We show that with few exceptions every local isometric automorphism of the group algebra $L^p(G)$ of a compact group $G$ is an isometric automorphism., Comment: To appear in Proc. AMS