# Example of linear functionals on $B(H)$

Let $$H$$ be a Hilbert space and $$B(H)$$ denotes the space of all continuous linear operators on $$H$$. I am looking for a class/example of bounded linear functionals $$B(H)\to \mathbb C$$ which cannot be reduced to the type $$T\mapsto\sum_{i=1}^\infty k_i\left$$ for some fixed $$x_i$$'s and $$y_i$$'s in $$H$$ and $$k_i$$'s in $$\mathbb C$$. Do we have any explicit examples?

So, you are asking about non-normal functionals on $$B(H)$$. This is very similar to the question of what are the functionals on $$\ell_\infty$$ that are not in $$\ell_1$$?

Fix an ultrafilter $$U$$ on $$\mathbb N$$ and let $$(e_n)$$ be an ONB for $$H$$. Define

$$\langle \phi, T\rangle = \lim_{n,U} \langle Te_n, e_n\rangle\quad (T\in B(H)).$$

Then this functional is not normal.

• Thank you for answering. Can you suggest some literature where I could find these results along with proofs?
– NewB
Sep 29, 2020 at 22:14
• @NewB, I'd go for Takesaki, Theory of operator algebras, vol. 1. Sep 30, 2020 at 6:06
• As far as I could understand, you gave a recipe for linear functionals on B(H) which are not coming through the natural map $J:H^*\otimes^\gamma H\to B(H)^*$, since projective tensor product $H^*\otimes^\gamma H$ is nothing but trace class operators, the predual of $B(H)$. Does this work for any Banach space projective tensor product? I mean, can we get linear functionals on $(X\otimes^\gamma Y)^{*}$ which are not coming through J map, through the same recipe?
– NewB
Sep 30, 2020 at 15:47
• Also, do you mean non-principle ultrafilter? Because for principle ultrafilter generated by $k$, the functional defined above is $T\mapsto \left<Te_k,e_k\right>$, which is again normal. Am I missing here something?
– NewB
Oct 1, 2020 at 20:24
• @NewB, of course non-principal. Oct 2, 2020 at 7:08