C_n(Y) \simeq C_n(A) \otimes_A Y$ completely isometrically.

Ternary rings of operators
--------------------------

A *ternary ring of operators (TRO)* $M$ is a subspace of some ${{\mathcal{B}}}(H)$ such that $MM^*M \subseteq M$. It then follows that $M$ is an imprimitivity bimodule over $A = {\left[ MM^* \right]^{-\|\cdot\|}}$ and $B = {\left[ M^*M \right]^{-\|\cdot\|}}$. Every C\*-correspondence $X$ (on some $A$) is a TRO. However it gives an equivalence between ${{\mathcal{K}}}(X)$ and ${\left\langleX, X\right\rangle}$ which may differ from $A$ in principle. Moreover, for the TRO $M$ there are two nets $$\begin{aligned}
a_t =
\sum_{i=1}^{l_t} m_i^t (m_i^t)^*
{\quad\text{and}\quad}b_\lambda =
\sum_{i=1}^{k_\lambda } (n_i^\lambda)^* n_i^\lambda\end{aligned}$$ for $m_i^t, n_j^\lambda \in M$ such that all $[m_1^t, m_2^t, \dots, m_{l_t}^t]$ and $[(n_1^\la)^*, (n_2^\la)^*, \dots, (n_{k_\la}^\la)^*]$ are row contractions and $$\lim _t a_t m = m {\quad\text{and}\quad}\lim_\lambda m b_\lambda = m$$ for all $m\in M$; see for example the proof of [@BMP00 Theorem 6.1]. In particular the C\*-algebras $A = {\left[ MM^* \right]^{-\|\cdot\|}}$ and $B = {\left[ M^*M \right]^{-\|\cdot\|}}$ are $\si$-unital if and only if the nets $(a_t)$ and $(b_\la)$ can be chosen to be sequences; see [@Bro77 Lemma 2.3]. If any of the above happens then we will say that $M$ is a *$\si$-TRO*. These approximate identities provide an efficient 