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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?

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

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