non-existence of nonzero special linear map on $B(H)$

Question

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!

You might want to check out this: https://math.stackexchange.com/questions/860886/no-trace-on-bh-if-h-is-infinite-dimensional — Merry, Oct 23 '18 at 17:41

score 1 · Accepted Answer · answered Oct 23 '18 at 16:37

1

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.

answered Oct 23 '18 at 16:37

I'm curious what your argument is. One gets easily that $\phi$ is zero on infinite projections, but my own argument to conclude that $\phi$ is zero is a bit clumsy. – Martin Argerami Oct 23 '18 at 18:21
@Martin Argerami,I think your argument is very clear. – math112358 Oct 24 '18 at 03:29

non-existence of nonzero special linear map on $B(H)$

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