Let H be a complex Hilbert space and denote by B s (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 B s (H) which preserve the order ≤ in both directions.

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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 rankone idempotents of a Banach space which preserve zero products in both directions. # 2002 Elsevier Science (USA)

Abstract. We show that the existence of a surjective isometry (which is merely a distance preserving map) between the unitary groups of unital C * -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 C * -algebras. IntroductionThe study of linear isometries between function spaces or operator algebras has a long history dating back to the early 1930's. For an excellent comprehensive treatment of related results we refer to the two volume set [5,6]. The most fundamental and classical results of that research area are the Banach-Stone theorem describing the structure of all linear surjective isometries between the Banach spaces of continuous functions on compact Hausdorff spaces and its noncommutative generalization, Kadison's theorem [13], which describes the structure of all linear surjective isometries between general unital C * -algebras. One immediate consequence of those results is that if two C * -algebras are isometrically isomorphic as Banach spaces, then they are isometrically isomorphic as Jordan *-algebras, too. This provides a good example of how nicely the different sides (in the present case the linear algebraic -geometrical structure and the full algebraic, more precisely, Jordan *-algebraic structure) of one complex mathematical object may be connected to or interact with each other. We mention another famous result of similar spirit which also concerns isometries. This is the celebrated Mazur-Ulam theorem stating that any surjective isometry between normed real linear spaces is automatically an affine transformation (hence equals a real linear surjective isometry followed by a translation). This means that if two normed real linear spaces are isometric as metric spaces, then they are isometrically isomorphic as normed linear spaces, too.2010 Mathematics Subject Classification. 47B49, 46J10.

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 ∈ N fixed.