1. A new look at local maps on algebraic structures of matrices and operators.
- Author
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Molnár, Lajos
- Subjects
- *
UNITARY groups , *AUTOMORPHISM groups , *FUNCTION algebras , *HILBERT space , *MATRICES (Mathematics) , *AUTOMORPHISMS - Abstract
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]
- Published
- 2022