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

We study maps between positive definite or positive semidefinite cones of unital \(C^*\) $C^{\ast }$ -algebras. We describe surjective maps that preserve (1)

the norm of the quotient or product of elements;

(2)

the spectrum of the quotient or product of elements;

(3)

the spectral seminorm of the quotient or product of elements.

These maps relate to the Jordan \(*\) $\ast$ -isomorphisms between the specified \(C^*\) $C^{\ast }$ -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 \(C^*\) $C^{\ast }$ -algebras, the other types of surjections can be extended to a Jordan \(*\) $\ast$ -isomorphism or a Jordan \(*\) $\ast$ -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 \(C^*\) $C^{\ast }$ -algebras. Applying it, we provide positive solutions to the problem posed by Molnár for general unital \(C^*\) $C^{\ast }$ -algebras.