Non-linear characterization of Jordan \(*\)-isomorphisms via maps on positive cones of \(C^*\)-algebras

Abstract

We study maps between positive definite or positive semidefinite cones of unital \(C^*\)-algebras. We describe surjective maps that preserve

1. (1)

   the norm of the quotient or product of elements;

2. (2)

   the spectrum of the quotient or product of elements;

3. (3)

   the spectral seminorm of the quotient or product of elements.

These maps relate to the Jordan \(*\)-isomorphisms between the specified \(C^*\)-algebras. While a surjection between positive definite cones that preserves the norm of the quotient of elements may not be extended to a linear map between the underlying \(C^*\)-algebras, the other types of surjections can be extended to a Jordan \(*\)-isomorphism or a Jordan \(*\)-isomorphism followed by 2-sided multiplication by a positive invertible element. We also study conditions for the centrality of positive invertible elements. We generalize “the corollary” regarding surjections between positive semidefinite cones of unital \(C^*\)-algebras. Applying it, we provide positive solutions to the problem posed by Molnár for general unital \(C^*\)-algebras.