Certain properties of derivations

Spivack, Mark

doi:10.1090/S0002-9939-1987-0877045-4

Certain properties of derivations
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by Mark Spivack

Proc. Amer. Math. Soc. 99 (1987), 712-718

DOI: https://doi.org/10.1090/S0002-9939-1987-0877045-4

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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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