Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}I⊆\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? $\mathbb{R}I=\Phi(\mathbb{R}I)$(here I is the identity operator)
If yes how can i prove that and if no what is an example that rejects this claim?
So thanks for your help.
The exact same idea as in the answer to your other question works. That is, now take a Hamel basis of $B(H)$ that extends a Hamel basis of $\mathbb RI$, and you can still get a $\mathbb Q$-linear map (so additive) such that $\mathbb RI\subsetneq \Phi(\mathbb RI)$.