pass to an appropriate Hilbert subspace where $\pi$ acts non-degenerately. This will always be the case in the current paper.

Tensor products
---------------

Let us recall the following results of Blecher [@Ble97] concerning tensor products. For this subsection let us fix a right $A$-module $X$ and a C\*-correspondence ${}_A Z_B$. We further assume that $Z$ is non-degenerate; otherwise all that follow hold for the essential part ${\left[ \phi_Z(A)Z \right]^{-\|\cdot\|}}$ of $Z$.

Blecher [@Ble97 Theorem 3.1] has shown that the right Hilbert modules are asymptotic summands of the free modules $C_n(A) = \sum_{i=1}^n A$, i.e. for $X_A$ there are completely contractive $A$-module maps $$\phi_\al \colon X \to C_{n(\al)}(A) {\quad\text{and}\quad}\psi_\al \colon C_{n(\al)}(A) \to X$$ such that $\psi_\al \phi_\al \to {{\operatorname{id}}}_X$ strongly. One of the main consequences [@Ble97 Theorem 4.3] is that the stabilized Haagerup tensor product $X \otimes_A^h Y$ is completely isometrically isomorphic to the stabilized Hilbert-module tensor product $X \otimes_A Y$. This is derived by following the diagonals in the diagram $$\xymatrix@R=5mm@C=20mm{
X \otimes_A^h Y \ar@{-->}[dr]_{\phi_\al} \ar[rr]^{{{\operatorname{id}}}} \ar@{-->}[dd] & & X \otimes_A^h Y \\
& C_{n(\al)}(Y) \ar[ur]_{\psi_\al} \ar@{-->}[dr]^{\psi_\al} & \\
X \otimes_A Y \ar[ur]^{\phi_\al} \ar[rr]^{{{\operatorname{id}}}} & & X \otimes_A Y \ar[uu]
}$$ and using the fact that $C_n(A) \otimes_A^h Y \simeq 