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Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
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Bounded operators $T: B(H)\to H$ whose Kernel is a Lie algebra
The answer is yes :
Choose injective $A \in B(H)$ and $h \in H$ such that $h$ and $Ah$ are linear independent and define $\phi(T) = A T h$.
Then it is easy to see that there doesn't exist a $k \in H …