Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Certain properties of derivations


Author: Mark Spivack
Journal: Proc. Amer. Math. Soc. 99 (1987), 712-718
MSC: Primary 47D25
MathSciNet review: 877045
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Abstract: We consider two properties of implemented derivations on operator algebras, and give applications. One provides a simple test and leads to examples of nonimplemented derivations on commutative algebras. The other is stronger and yields a necessary and sufficient condition for derivations on $ pB(H){p^ \bot }$ to be implemented, where $ H$ is a Hilbert space and $ p$ is a projection on $ H$. Any algebra $ S$ on $ H$ has an extension to an algebra $ {S_2}$ acting on $ H \oplus {\mathbf{C}}$ containing such an algebra. We show that any derivation $ \delta $ on an algebra $ S$ is implemented if and only if $ \delta $ has a bounded strongly continuous extension to $ {S_2}$. If so we can construct an implementing operator explicitly.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0877045-4
Article copyright: © Copyright 1987 American Mathematical Society