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 $vin M(B_0(H)otimes A)$ 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 $(xotimes 1)v_1 =v_2(x otimes 1).$

**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 $Asubseteq B(K)$ 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 $B(H_1oplus H_2)$, we have canonical inclusions

$$B(H_1,H_2) otimes A subseteq B(H_1oplus H_2) otimes Asubseteq M(B_0(H_1 oplus H_2) otimes A)$$

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 $M(B_0(H_1oplus H_2)otimes A)$.

However, also the above does not seem quite satisfactory.