Non-linear characterization of Jordan $\ast$-isomorphisms via maps on positive cones of $C^{\ast }$-algebras

Abstract

We study maps between positive definite or positive semidefinite cones of unital $C^{\ast }$-algebras. We describe surjective maps that preserve (1) the norm of the quotient or multiplication of elements; (2) the spectrum of the quotient or multiplication of elements; (3) the spectral seminorm of the quotient or multiplication of elements. These maps relate to the Jordan $\ast$-isomorphisms between the specified $C^{\ast }$-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 $C^{\ast }$-algebras, the other types of surjections can be extended to a Jordan $\ast$-isomorphism or a Jordan $\ast$-isomorphism followed by the implementation 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 $C^{\ast }$-algebras. Applying it, we provide positive solutions to the problem posed by Molnár for general unital $C^{\ast }$-algebras.