I am trying to understand the proof of Theorem $9.4.1$ from the book titled Operator spaces by Effros Ruan.
Let A and $B$ be unital $C^{\ast}$-algebras then a linear functional $F: A \otimes^h B \to \mathbb{C}$ is bounded if and only if there exist unital $^{\ast}$-representations $\Pi_A: A\to B(H)$ and $\Pi_B : B \to B(K)$, vector $\zeta \in H$ and $\eta \in K$ and a bounded linear operator $T: K \to H$ such that $F( a \otimes b)=\langle \Pi_A(a)T\Pi_B(b) \eta, \zeta \rangle_H$
The proof goes as follows
If $F: A \otimes B \to \mathbb{C}$ is a linear functional satisfying the given condition, then for any $a=[a_1,...,a_r] \in M_{1,r}(A)$ and $b=[b_1,...,b_r]^T \in M_{r,1}(B)$ we have
\begin{align*} \vert F( a \odot b) \vert &= \vert \sum F(a_i \otimes b_i) \vert \\ &= \vert \sum \langle \Pi_A(a_i)T \Pi_B(b_i)\eta, \zeta \rangle \vert \\ &=\vert \langle (\Pi_A)_{1,r}(a)(I_r \otimes T)(\Pi_B)_{r,1}(b)\eta , \zeta \rangle \vert \end{align*}
Can someone please explain me how does one obtains the last line ?
The last line is a condensed way of writing \begin{align*} \sum_j\Pi_A(a_j)T\Pi_B(b_j) &=\begin{bmatrix} \Pi_A(a_1)&\cdots &\Pi_A(a_r)\end{bmatrix} \begin{bmatrix} T\\ & \ddots\\ && T\end{bmatrix} \begin{bmatrix} \Pi_B(b_1)\\ \vdots\\ \Pi_B(b_r)\end{bmatrix}\\[0.3cm] &=(\Pi_A)_{1,r}(a)\,\begin{bmatrix} T\\ & \ddots\\ && T\end{bmatrix} (\Pi_B)_{r,1}(b). \end{align*}