{{\mathcal{L}}}(X,Y)$. If $v$ is further a left module map then so is $u$. Therefore we now restrict our attention just to right Hilbert modules. We will use the following lemma.

\[L: cenv linking\] Let $X_A$ be a right Hilbert module. Then we have that $${\mathrm{C}^*_{\textup{env}}}(\begin{bmatrix} A & 0 \\ X & 0 \end{bmatrix}) = {{\mathfrak{L}}}(X) .$$

For simplicity let us write ${{\mathcal{A}}}(X)$ for the operator algebra $\begin{bmatrix} A & 0 \\ X & 0 \end{bmatrix}$. It is clear that ${{\mathfrak{L}}}(X)$ is a C\*-cover of ${{\mathcal{A}}}(X)$. Therefore there exists a unique \*-epimorphism $\Phi \colon {{\mathfrak{L}}}(X) \to {\mathrm{C}^*_{\textup{env}}}({{\mathcal{A}}}(X))$ such that $$\Phi(\begin{bmatrix} a & 0 \\ x & 0 \end{bmatrix}) = i(a) + i(x) {\text{ for all }}a \in A, x \in X$$ for the embedding $i \colon {{\mathcal{A}}}(X) \to {\mathrm{C}^*_{\textup{env}}}({{\mathcal{A}}}(X))$. Let $H$ be a Hilbert space where ${\mathrm{C}^*_{\textup{env}}}({{\mathcal{A}}}(X))$ acts non-degenerately.

Since $A$ acts non-degenerately on the right of $X$ and ${{\mathcal{K}}}(X)$ acts non-degenerately on the left of $X$ we can choose a c.a.i. for ${{\mathfrak{L}}}(X)$ of the form $(a_i \oplus k_i)$ for a c.a.i. $(a_i)$ of $A$ and a c.a.i. $(k_i)$ for ${{\mathcal{K}}}(X)$. Since $\Phi$ is surjective then $(\Phi(a_i) + \Phi(k_i))$ is a c.a.i. for ${\mathrm{C}^*_{\textup{env}}}({{\mathcal{A}}}(X))$. Write $$P := \text{sot-}\lim \Phi(a_i) {\quad\text{and}\quad}Q 