by a TRO $M$. Then for every non-degenerate (injective) representation $(\pi, t)$ of ${}_A E_A$ on a Hilbert space $K$ there exists a non-degenerate (resp. injective) representation $(\si, s)$ of ${}_B F_B$ on a Hilbert space $H$ and a TRO-representation $\phi$ of $M$ in ${{\mathcal{B}}}(H, K)$ such that $${\left[ t(E) \right]^{-\|\cdot\|}} = {\left[ \phi(M) s(F) \phi(M)^* \right]^{-\|\cdot\|}}$$ and $${\left[ s(F) \right]^{-\|\cdot\|}} = {\left[ \phi(M)^* t(E) \phi(M) \right]^{-\|\cdot\|}}.$$ If $(\pi,t)$ admits a gauge action then so does $(\si,s)$.

To avoid technical notation we show the dual statement. That is, given a non-degenerate (injective) representation $(\si, s)$ of ${}_B F_B$ acting on $H$, we will construct the required $(\pi, t)$ and $\phi$. Let $K = M \otimes_B H$ and define the representation $$\phi \colon M \to {{\mathcal{B}}}(H, K) \text{ such that } \phi(x)\xi = x \otimes \xi .$$ Then $\phi$ is a TRO representation of $M$. Furthermore we have an induced non-degenerate (resp. injective) representation $$\pi \colon A \to {{\mathcal{B}}}(K) \text{ such that } \pi(a) x \otimes \xi = (ax) \otimes \xi.$$ Since $E \simeq M \otimes_B F \otimes_B M^*$ we can define $$t \colon E \to {{\mathcal{B}}}(K) \text{ such that } t(m \otimes f \otimes n^*) x \otimes \xi = m 