shown that $J \subseteq J_X$ if and only if $A \hookrightarrow {{\mathcal{O}}}(J,X)$ if and only if ${{\mathcal{T}}}_X^+ \hookrightarrow {{\mathcal{O}}}(J, X)$. The proof follows by [@KatKri06] and the diagram $$\begin{aligned}
\xymatrix@R=1em@C=4em{ {{\mathcal{T}}}_X \ar[rr] \ar[dr] & & {{\mathcal{O}}}_X\\
&  {{\mathcal{O}}}(J, X) \ar[ur] &
}\end{aligned}$$ where the arrows indicate canonical $*$-epimorphisms. Therefore the $J$-relative algebras for $J \subseteq J_X$ are the only Pimsner algebras that contain an isometric copy of the C\*-correspondences. Beyond this point we “lose” information of the original data.

We say that a representation $(\pi,t)$ admits a gauge action $\{\be_z\}_{z \in {\mathbb{T}}}$ if every $\be_z$ is an automorphism of ${\mathrm{C}^*}(\pi,t)$ such that $$\label{eq:ga}
\be_z(\pi(a)) = \pi(a) {\quad\text{and}\quad}\be_z(t(x)) = z t(x)$$ for all $a \in A$ and $x \in X$, and the family $\{\be_z\}_{z \in {\mathbb{T}}}$ is point-norm continuous. Since ${\mathrm{C}^*}(\pi,t)$ is densely spanned by the monomials of the form $t(x_1)\cdots t(x_n) t(y_m)^* \cdots t(y_1)^*$, an ${\varepsilon}/3$-argument implies that if $\{\be_z\}_{z \in {\mathbb{T}}}$ is a family of $*$-homomorphisms of ${\mathrm{C}^*}(\pi,t)$ that satisfies equation (\[eq:ga\]) then it is point-norm continuous.

The Gauge-Invariant-Uniqueness-Theorem (GIUT) is a fundamental result for lifting representations of the C\*-correspondence to operator algebras. This type of result was initiated by an Huef and Raeburn for Cuntz-Krieger algebras [@HueRae97 Theorem 