Constructing intertwiners between representations of compact quantum groups Consider the following paper by Van Daele en Maes Notes on compact quantum groups. For convenience of the reader, here is a picture of the relevant section:

(1) How is compact operator defined in this context? For example, what is meant with $x$ is a compact operator from $H_1 \to H_2$? Is this the usual definition of compactness? I.e. the image of the unit ball is precompact. Or is something else going on?
(2) In the proof, one considers the object $\mathcal{B}_0(\mathcal{H}_1, \mathcal{H}_2)\otimes A$. How is this tensor product defined in this context? Surely $\mathcal{B}_0(\mathcal{H}_1, \mathcal{H}_2)$ is no $C^*$-algebra so this is not a tensor product of $C^*$-algebras.
 A: Yes, an operator is compact if it maps the unit ball into a precompact set.  This is the usual definition for Banach spaces.  For Hilbert spaces the same applies (see e.g. these notes). Indeed, a bounded linear $T:H_1\rightarrow H_2$ is compact if and only if $T^*T$ is compact, iff $TT^*$ is compact, iff $T^*$ is compact.

To deal with the second question, I think one proceeds as follows.  Firstly, observe that there is a $*$-algebra isomorphism
$$ \mathcal B(H_1\oplus H_2) \cong \begin{pmatrix}
\mathcal B(H_1) & \mathcal B(H_2, H_1) \\
\mathcal B(H_1, H_2) & \mathcal B(H_2)
\end{pmatrix}. $$
To see this, think about the matrix acting on $H_1\oplus H_2$ written as a column vector.  Under this isomorphism, compact operators behave as you might hope,
$$ \mathcal K(H_1\oplus H_2) \cong \begin{pmatrix}
\mathcal K(H_1) & \mathcal K(H_2, H_1) \\
\mathcal K(H_2, H_1) & \mathcal K(H_2)
\end{pmatrix}. $$
Thus I can speak of $\mathcal K(H_1,H_2)$ as a "corner" of $\mathcal K(H_1\oplus H_2)$.
So, one can define $\mathcal K(H_1,H_2) \otimes A$ as the closure of $\mathcal K(H_1,H_2) \odot A$ inside $\mathcal K(H_1\oplus H_2) \otimes A$.  Things are nicer than this: let $p_i$ be the projection of $H_1\oplus H_2$ onto $H_i$.  Then $p_2 \mathcal K(H_1\oplus H_2) p_1$ is isomorphic to $\mathcal K(H_1,H_2)$, and $p_i\otimes 1\in M(\mathcal K(H_1\oplus H_2)\otimes A)$.  One can check that $\mathcal K(H_1,H_2) \otimes A$ is isomorphic to $(p_2\otimes 1)(\mathcal K(H_1\oplus H_2)\otimes A)(p_1\otimes 1)$.

The notes by Maes and Van Daele are nice, but I do find that there are various little inaccuracies, or points like this which are not (well) explained.  The original papers by Woronowicz are terse, but I think a pleasure to read, and will improve your intuition about the subject.  You could also look at the book of Timmermanns, but that takes a different approach.

In a comment, Ruy points our how to get this tensor product via representing on a Hilbert space.
An approach using more theory would be to use Hilbert $C^\ast$-modules (I follow chapter 4 of Lance's book).  $\mathcal K(H_1, H_2)$ is a right module over $\mathcal K(H_1)$ for the "inner-product" $(S|T) = S^*T$.  $A$ is a module over itself.  The exterior tensor product of modules gives $\mathcal K(H_1, H_2) \otimes A$ as a right module over $\mathcal K(H_1)\otimes A$.  A rather tedious check shows that the norms of all three approaches are the same.
For the application, we need:

*

*For a state $h\in A^*$ we want to make sense of $\iota\otimes h$ as a map $\mathcal K(H_1,H_2)\otimes A \rightarrow \mathcal K(H_1,H_2)$;

*For a $*$-homomorphism $\Phi:A\rightarrow A\otimes A$ we need to make sense of $\iota\otimes\Phi$ as a homomorphism (suitable interpreted) $\mathcal K(H_1,H_2)\otimes A \rightarrow \mathcal K(H_1,H_2)\otimes A \otimes A$.

I think my original approach, of viewing things as a "corner" of $C^\ast$-algebra, is probably the easiest way to get these properties.
