Maps on Positive Cones of C^(*)-algebras

We prove that a surjective map(on the positive cones of unital C^(*)-algebras)preserves the minimum spectrum values of harmonic means if and only if it has a Jordan *-isomorphism extension to the whole algebra.We represent weighted geometric mean preserving bijective maps on the positive cones of prime C^(*)-algebras in terms of Jordan *-isomorphisms of the algebras.We also characterize order isomorphisms and orthoisomorphisms of the projection lattice of the von Neumann algebra of all bounded linear operators on a Hilbert space,answering an open question arisen by Dye.Finally,we give a description for Fuglede-Kadison determinant preserving maps on the positive cone of a finite von Neumann algebra and improve Gaal and Nayak’s work on this topic.