Tensor product of operator values weights (in the theory of locally compact quantum groups)

Question

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.

Answer 1

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.

Answer 2

Each normal weight is a supremum of normal positive functionals (second volume of Takesaki, theorem 1.11 in Chapter VII). For a normal positive functional $\phi$ you can just define $(\iota \otimes \phi)(z)$ as an element of $M_{+}$, hence a nice continuous linear function on the positive part of the predual of $M$. A supremum of continuous functions is lower semicontinuous, so indeed $\iota \otimes \psi$ is an element of the extended positive part of $M$. Normality of $\iota \otimes \psi$ also follows from representing $\psi$ as a supremum of normal positive functionals.