# Example of a ternary Lie ideal which is not a Lie ideal

Let $$H$$ and $$K$$ be Hilbert spaces and $$V\subset B(H,K)$$ be a ternary ring of operators i.e. $$xy^*z \in V$$ for all $$x,y,z \in V$$. Let $$I$$ be a closed subspace of $$V$$. $$I$$ is called a ternary Lie ideal of $$V$$ provided $$\operatorname{span}\{ab^*c-cb^*a: a,c \in V, b\in I \} \subset I$$. One can check that every self-adjoint closed ternary Lie ideal of a $$C^{\ast}$$-algebra is actually a Lie ideal.

Does there exist an example of a ternary Lie ideal of a $$C^{\ast}$$-algebra which is not a Lie ideal?

I cannot see any example. Any ideas?

• What do you mean by a Lie ideal in case $H\neq K$? – Ruy Mar 28 at 22:08
• @Ruy: Sorry I'm not aware of concept of Lie ideal in TROs. – Math Lover Mar 29 at 0:41