an additive bijective map

Question

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.

Answer 1

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 $\mathbb RI\subsetneq \Phi(\mathbb RI)$.