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On notion of asymptotic derivations
Author(s) -
Heo Jaeseong
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201010003
Subject(s) - mathematics , automorphism , von neumann algebra , limit (mathematics) , pure mathematics , simple (philosophy) , commutative property , zero (linguistics) , asymptotic analysis , von neumann architecture , unital , algebra over a field , mathematical analysis , philosophy , linguistics , epistemology
In this paper we introduce some notion of asymptotic derivations of a C *‐ and W *‐dynamical systems which naturally arises from a one‐parameter group of automorphisms. We show that an asymptotic derivation on a unital simple C *‐algebra or a von Neumann algebra is asymptotically inner. Every asymptotic derivation on finite dimensional C *‐algebras is induced by an inner derivation. We prove that any asymptotic derivation on commutative von Neumann algebras have a strong limit zero. An asymptotic derivation on the hyperfinite II 1 ‐factor given by some one‐parameter group of automorphisms can be induced by a (inner) derivation. Finally, we show that every asymptotic Jordan derivation of a C *‐dynamical system is an asymptotic derivation and is also induced by a derivation if there exists a limit.