Let $H$ be a Hilbert space. Working on locally compact quantum groups, I have met with operator of the form $\iota \otimes \omega_{\xi, \eta}$ with $\xi,\eta \in H$ and $\omega_{\xi,\eta}(T) = (T\xi,\eta)$. These operators should be defined on $B(H \otimes H)$ but I can't find their definitions.
First, I thought we could just define them on elementary tensor $A \otimes B$ with $A,B \in B(H)$ but the linear span of these elements is NOT norm dense in $B(H \otimes H)$, it is only $\sigma$-weakly dense in it (as usual for Von Neumann algebras, we can replace $\sigma$-weakly with $*$-strongly etc) so we can't use some BLT theorem to extend it to the whole space.
Secondly, I thought that it was define by the formula $$\big((\iota \otimes \omega_{\xi,\eta})(T)g, f\big) = (T(f \otimes \xi) , g \otimes \eta)$$ for all $f,g \in H$ since this formula is true if we let $T = A \otimes B$ with $A,B \in B(H)$ and if we use the naive definition. I think this would make sense as it would clearly be $\sigma$-weakly continuous but still, I can't find any document that clearly defines it.
In case it is not obvious, my question is : is my second definition the usual one for these operators? If not, what is the usual definition ?
Thanks in advance.
The terms you are looking for are slice map, partial trace, Fubini map; there is not universal name.
Your $\omega_{\xi,\eta}$ is an example of a normal linear functional. They are of the form $\def\tr\operatorname{Tr}$ $T\longmapsto \tr(AT)$, where $A\in \mathcal T(H)$, the trace-class operators.
One defines $(\iota\otimes\varphi)(T)$ as the unique $S\in B(H)$ such that $$\tr(AS)=\tr((A\otimes B)T),\qquad\qquad A\in\mathcal T(H),$$ where $\varphi=\tr(B\cdot)$. Such $S$ exists, because if we consider the map $\Gamma:\mathcal T(H)\to\mathbb C$ given by $\Gamma(A)=\tr((A\otimes B)T)$, this is a bounded linear functional on $\mathcal T(H)$, and as $B(H)$ is the dual of $\mathcal T(H)$, there exists $S\in B(H)$ such that $\Gamma(A)=\tr(SA)$ .
When $T=T_1\otimes T_2$ and $S=(\iota\otimes\varphi)(T_1\otimes T_2)$, we have $$ \tr(AS)=\tr((A\otimes B)(T_1\otimes T_2))=\tr(AT_1)\tr(BT_2)=\varphi(T_2)\tr(AT_1)=\tr\big(A[\varphi(T_2)T_1]\big) $$ for all $A$, and so $S=\varphi(T_2)T_1$. When $\varphi=\omega_{\xi,\eta}$, this gives your $$ (\iota \otimes \omega_{\xi,\eta}(T)g \mid f) = (T(f \otimes \xi) \mid g \otimes \eta). $$