I saw an exercise :There is no nonzero linear map $\phi:B(H) \rightarrow \mathbb{C}$ satisfying $\phi(ab)=\phi(ba)$ when $dim(H)=\infty$. I have no idea.Can anyone give me some hints,thanks!
2026-03-26 04:31:16.1774499476
non-existence of nonzero special linear map on $B(H)$
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This is because $B(H)$ is infinite, for $H$ infinite dimensional.
Take two isometries $S_0,S_1 \in B(H)$ such that $S_0S_0^* + S_1S_1^* = 1$. Now see what happens if you apply your map $\phi$ to that equation.