Let $M$ be a von Neumann algebra and $\psi$ a normal faithful semifinite weight on $M$. Then one should be able to form the object
$$\iota \otimes \psi: (M\overline{\otimes} M)\_+ \to \widehat{M\_+}.$$
This is for example used in the theory of locally compact quantum groups (in the sense of Vaes-Kustermans). I have been told that this is an "operator valued weight". Takesaki's second book contains a section about operator-valued weights, but I cannot find the definition of tensor product of operator-valued weights in the book.


Concretely, my question is: how to define the tensor product of operator-valued weights? References are appreciated.



Tensor product of operator values weights (in the theory of locally compact quantum groups)
The approach I had in mind is the following. For a reference, see Section 4, Chapter IX of Takesaki 2. The extended positive part $\widehat{M\_+}$ is by definition the space of positive-homogeneous, additive, lower semi-continuous maps $M\_\*^+\rightarrow [0,\infty]$. Given $x\in (M\bar\otimes M)\_+$ define $(\iota\otimes\varphi)(x) \in \widehat{M\_+}$ to be the map
$$ \omega \mapsto \varphi\big( (\omega\otimes\iota)(x) \big). $$
For $\omega\in M\_\*^+$, as $x$ is positive, also $(\omega\otimes\iota)(x)$ is positive, and so we obtain a well-defined member of $[0,\infty]$. Clearly $(\iota\otimes\varphi)(x)$ is positive-homogeneous and additive. If $(\omega\_i)$ is a net in $M\_\*^+$ converging to $\omega$ in norm, then $(\omega\_i\otimes\iota)(x) \rightarrow (\omega\otimes\iota)(x)$ $\sigma$-weakly. As $\varphi$ is $\sigma$-weakly lower semi-continuous (see Theorem 1.11 in Chapter VII of Takesaki) it follows that $\lim\_i (\iota\otimes\varphi)(x)(\omega\_i) \geq (\iota\otimes\varphi)(x)(\omega)$. Thus $(\iota\otimes\varphi)(x)$ is lower semi-continuous.


Clearly the map $\iota\otimes\varphi$ is positive-homogeneous and additive. Given $a\in M, \omega\in M\_\*^+$,
$$ (\iota\otimes\varphi)((a\otimes 1)^\*x(a\otimes 1))(\omega)
= \varphi\big( (a \omega a^\* \otimes\iota)(x) \big)
= (\iota\otimes\varphi)(x)(a \omega a^\*)
= \big( a^\* (\iota\otimes\varphi)(x) a \big)(\omega). $$
Thus $(\iota\otimes\varphi)((a\otimes 1)^\*x(a\otimes 1)) = a^\* (\iota\otimes\varphi)(x) a$.


Finally, if $(x\_i)$ increases to $x$ in $(M\bar\otimes M)\_+$ then for each $\omega\in M\_\*^+$ we have that $(\omega\otimes\iota)(x\_i)$ increases to $(\omega\otimes\iota)(x)$ so by normality of $\varphi$, it follows that $(\iota\otimes\varphi)(x\_i)(\omega)$ increases to $(\iota\otimes\varphi)(x)(\omega)$. Hence $(\iota\otimes\varphi)(x\_i)$ increases to $(\iota\otimes\varphi)(x)$. So $\iota\otimes\varphi$ is normal. So $\iota\otimes\varphi$ is an operator-valued weight.


