On Maps Preserving the Spectrum on Positive Cones of Operator Algebras

We consider the spectrum-preserving maps on positive definite cones of C*-algebras or von Neumann algebras. We first introduce some basic properties of Jordan isomorphism. Then, we study the additively spectrum preserving property and the multiplicatively spectrum-preserving property, and prove that these maps can be characterized by Jordan isomorphisms between C*-algebras.