Characterizations of Jordan *-isomorphisms of C⁎-algebras by weighted geometric mean related operations and quantities

Abstract

In this paper we consider three operations on positive definite cones of C⁎-algebras which are related to weighted geometric means and appear in the formulas defining various versions of quantum Rényi relative entropy. We show how Jordan *-isomorphisms between C⁎-algebras can be characterized by the preservation of the norms of products under those operations or by the preservation of the operations themselves. We also obtain conditions for the commutativity of the underlying algebras by showing that we have that property if one of the quantities under considerations can be transformed by a surjective map to another different such quantity.