Transformations preserving the norm of means between positive cones of general and commutative $C^*$-algebras

Abstract

In this paper, we consider a (nonlinear) transformation Φ of invertible positive elements in C * -algebras which preserves the norm of any of the three fundamental means of positive elements; namely, Φ(A)mΦ(B) = AmB , where m stands for the arithmetic mean A∇B = (A + B)/2, the geometric meanAssuming that Φ is surjective and preserves either the norm of the arithmetic mean or the norm of the geometric mean, we show that Φ extends to a Jordan * -isomorphism between the underlying full algebras. If Φ 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 (X) ⊂ C 0 (X) + → C 0 (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 (X) contains sufficiently many elements to peak on compact G δ sets. When the image T (F (X)) also contains sufficiently many elements to peak on compact G δ sets, T extends to an algebra * -isomorphism between the underlying full function algebras.