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 φ 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 α_V,W of the unitary group such that φ(V)φ(W)=α_V,W(VW), then either φ itself or -φ 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.

The author acknowledges supports from the Ministry for Innovation and Technology, Hungary, grant NKFIH-1279-2/2020 and from the National Research, Development and Innovation Office of Hungary, NKFIH, Grant No. K115383, K134944.