Before asking my question, let me introduce the relevant terminology.


Throughout, let $(A, \Delta)$ be a compact quantum group.


**Definition:** A representation $v$ on the Hilbert space $H$ is an element $v\in M(B\_0(H)\otimes A)$ such that $(\text{id}\otimes \Delta)(v) = v\_{(12)}v\_{(13)}$. Here the subscripts with the brackets denote the leg numbering notation.


**Definition:** An intertwiner from the representation $(H\_1, v\_1)$ to the representation $(H\_2, v\_2)$ is an element in $B(H\_1,H\_2)$ such that $(x\otimes 1)v\_1 =v\_2(x \otimes 1).$


**Question**: How should the multiplications $(x \otimes 1) v\_1$ and $v\_2 (x \otimes 1)$ be interpreted?


One way of making sense of these multiplications is as follows:


Let $A\subseteq B(K)$ be a faithful and non-degenerate representation, say the universal GNS representation of $A$. Then we have a canonical inclusion $M(B\_0(H\_1) \otimes K) \subseteq B(H\_1 \otimes K)$ and we can interpret $x \otimes 1$ as an operator $H\_1 \otimes K \to H\_2 \otimes K$ and $v\_1$ as an operator $H\_1 \otimes K \to H\_1 \otimes K$ and we can simply form the composition $(x \otimes 1)v\_1$ in $B(H\_1 \otimes K, H\_2 \otimes K)$. Similarly for the other side.


However, can one give a definition that is "space-free", i.e. does not refer to a choice of faithful and non-degenerate representation?


EDIT: Maybe the following works:


Viewing $B(H\_1, H\_2)$ as a corner of $B(H\_1\oplus H\_2)$, we have canonical inclusions
$$B(H\_1,H\_2) \otimes A \subseteq B(H\_1\oplus H\_2) \otimes A\subseteq M(B\_0(H\_1 \oplus H\_2) \otimes A)$$
and we also have a canonical inclusion
$$M(B\_0(H\_1)\otimes A)\subseteq M(B\_0(H\_1 \oplus H\_2)\otimes A)$$ so we can perform the multiplication in $M(B\_0(H\_1\oplus H\_2)\otimes A)$.


However, also the above does not seem quite satisfactory.



Definition intertwiner of representations of compact quantum groups
Perhaps what you are after is the language of Hilbert $C^\ast$-modules. Here I follow Lance's book (if someone knows a good online reference, please add a comment!) For a $C^\ast$-algebra $A$ and a Hilbert space $H$ we consider the right $A$-module $H\odot A$ with $A$-valued inner-product
$$ \big( \xi\otimes a \big| \eta\otimes b \big) = (\xi|\eta) a^\*b. $$
A little work is required to show that this is positive definitive, see Lance page 6. Let $H\otimes A$ be the completion.


It is not so hard to show that the algebra of "compact" adjointable operators $\newcommand{\mc}{\mathcal}\mc K(H\otimes A)$ is $\*$-isomorphic to $\mc K(H)\otimes A$, see Lance page 10. Thus the algebra of all adjointable operators satisfies
$$ \mc L(H\otimes A) \cong M(\mc K(H\otimes A)) \cong M(\mc K(H)\otimes A). $$
Notice the RHS is exactly the algebra which interests us!


Given a bounded linear map $T:H\_1\rightarrow H\_2$ it is clear that $T\otimes\iota$ is a bounded linear map $H\_1\otimes A\rightarrow H\_2\otimes A$ which has adjoint $T^\*\otimes\iota$. Thus, just using composition of adjointable maps, for $V\_1\in\mc L(H\_1\otimes A)$ and $V\_2\in\mc L(H\_2\otimes A)$ we have that
$$ (T\otimes\iota)V\_1, \quad V\_2(T\otimes\iota) \in \mc L(H\_1\otimes A, H\_2\otimes A).
$$
This gives the precise meaning you are after.




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This does agree with the other interpretation. I suppose the way to see this is to consider the isomorphism
$$ \mc L((H\_1\oplus H\_2)\otimes A) \cong
\begin{pmatrix} \mc L(H\_1\otimes A) & \mc L(H\_2\otimes A, H\_1\otimes A) \\
\mc L(H\_1\otimes A, H\_2\otimes A) & \mc L(H\_2\otimes A)
\end{pmatrix}. $$
This is proved by using the isomorphism $(H\_1\oplus H\_2)\otimes A \cong (H\_1\otimes A) \oplus (H\_2\otimes A)$. Now chase all the isomorphisms through.


