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The primary aim of this paper is to present the complete description of the isomorphisms between positive definite cones of C^*-algebras with respect to the recently introduced Wasserstein mean and to show the nonexistence of nonconstant such morphisms into the positive reals in the case of von Neumann algebras without type I_2, I_1 direct summands. A comment on the algebraic properties of the Wasserstein mean relating associativity is also made. [ABSTRACT FROM AUTHOR]

In a very general setting, we introduce a new type of local maps, a new sort of reflexive closure of a given class of transformations relative to a given operation that we call operational reflexive closure, and a corresponding concept of reflexivity. We calculate the operational reflexive closures of some important classes of transformations and significantly strengthen former 2-reflexivity results concerning the automorphism groups of various operator structures. A typical new result is this: if ø is a map from the unitary group over a separable infinite dimensional Hilbert space into itself with the property that for any pair V, W of unitaries there is a group automorphism αV,W of the unitary group such that ø(V0 ø(W) = αV,W (VW), then either ø itself or ø is a group automorphism. This result substantially generalizes a former one on the 2-reflexivity of the automorphism group of the unitary group. We also present open problems and questions for further study. [ABSTRACT FROM AUTHOR]