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.
Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ (fg) of the product of any two elements f and g in A coincides with the spectrum σ (Tf T g). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and nonmultiplicative unital map from a commutative C*-algebra into itself such that σ (Tf T g) = σ (fg) holds for every f, g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ (Tf T g) ⊂ σ (fg) holds for every f, g.
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