& y_1^* \\ x_1 & u_1 \end{bmatrix}
\cdot
\begin{bmatrix} a_2 & y_2^* \\ x_2 & u_2 \end{bmatrix}
=
\begin{bmatrix} a_1a_2 + {\left\langley_1, x_2\right\rangle}  & (y_2 a_1^*)^* + (u_2^*(y_1))^* \\ x_1 a_2 + u_1(x_2) & \Theta_{x_1, y_2} + u_1 u_2 \end{bmatrix} .$$ The augmented linking algebra becomes a C\*-algebra over an operator norm when the matrices are seen to act on the right Hilbert module $A + X$. The *linking algebra* is defined as the C\*-subalgebra $${{\mathfrak{L}}}(X) = \begin{bmatrix} A & X^* \\ X & {{\mathcal{K}}}(X) \end{bmatrix}.$$ When $X$ is a C\*-correspondence over $A$ then it admits the operator bimodule structure by viewing the left action as the matrix multiplication $$\begin{bmatrix} 0 & 0 \\ \phi_X(a) x & 0 \end{bmatrix}
=
\begin{bmatrix} 0 & 0 \\ 0 & \phi_X(a) \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 \\ x & 0 \end{bmatrix}.$$

\[P: uneq=cc\] Let ${}_A X_A$ and ${}_B Y_B$ be C\*-correspondences. Then ${}_A X_A$ and ${}_B Y_B$ are completely isometric as operator bimodules if and only if ${}_A X_A$ and ${}_B Y_B$ are unitarily equivalent as C\*-correspondences.

The first observation is that if $v \in {{\mathcal{L}}}(X, Y)$ is invertible then the polar decomposition $v = u |v|$ gives a unitary $u \in 