Obtaining linear functionals on $B(H)$ using ultrafilters. Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i}\in \mathbb N$ and $X=H^*\otimes^\pi H$ is the projective tensor product. We have a natural isometry $$J:X\to X^{**}=B(H)^*$$ given by $J_{\sum k_1x_i\otimes y_i}(T)=\sum k_i\left<T(y_i),x_i\right>=\sum k_iT(x_i\otimes y_i)$ for all $T\in (H^*\otimes^\pi H)^*=B(H)$
I vaguely remember reading somewhere that $\psi:B(H)\to \mathbb C$ defined as $$\psi(T)=\lim_{n, U}\left<Te_n,e_n\right>$$ where $U$ is any is non-principle ultrafilter on $\mathbb N$ , is a well defined bounded linear functional which do not lie in the image of map $J$. How do we prove that $\psi$ is not in $J(X)$?
Further, is it correct to write $$\lim_{n, U}T(e_n\otimes e_n)=T(\lim_{n, U}e_n\otimes e_n)$$

$\psi$ vanishes on compact operators, while this is not the case for the members of $J(X)$.
Regarding the last question, I see no meaning in the RHS.
