1 results
On the Range Inclusion for Normal Derivations on C*-algebras
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 61 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 22 February 2017, pp. 1-11
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- Article
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For a von Neumann subalgebra
$A \, \subseteq \, {\cal B}({\cal H})$ and any two elements a, b ∈ A with a normal, such that the corresponding derivations da and db satisfy the condition ‖db(x)‖ ≤ ‖da(x)‖ for all x ∈ A, there exist completely bounded (a)ʹ-bimodule map 
$\varphi : {\cal B}({\cal H}) \rightarrow {\cal B}({\cal H})$ such that db|A = φ da|A=daφ|A. (In particular db(A) ⊆ da(A).) Moreover, if A is a factor, then φ can be taken to be normal and these equalities hold on 
${\cal B}({\cal H})$ instead of just on A. This result is not true for general (even primitive) C*-algebras 
${\cal A}$.
