 

Lajos Molnár
A new look at local maps on algebraic structures of matrices and operators
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Published: 
March 9, 2022. 
Keywords: 
Local maps, reflexive closures, automorphisms, derivations, operator algebras, matrix algebras, groups of operators and matrices, function algebras. 
Subject: 
Primary: 47B49, 46L40, 46L57. Secondary: 47B47, 47B65. 


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 2reflexivity 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 2reflexivity of the automorphism group of the unitary group.
We also present open problems and questions for further study.


Acknowledgements
The author acknowledges supports from the Ministry for Innovation and Technology, Hungary,
grant NKFIH12792/2020 and from the National Research, Development and Innovation Office of Hungary, NKFIH, Grant No. K115383, K134944.


Author information
Lajos Molnár:
Bolyai Institute
University of Szeged
Aradi vértanúk tere 1.
H6720 Szeged, Hungary, and
Department of Analysis, Institute of Mathematics
Budapest University of Technology and Economics
Muegyetem rkp. 3.
H1111 Budapest, Hungary
molnarl@math.uszeged.hu

