# oa.operator algebras – Definition intertwiner of representations of compact quantum groups

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.