I'm reading the book "Invitation to $C^*$-algebras" by Arveson and I'm focused on understanding the summation as defined here:
Can someone explicitely describe what the map $\sum_i \pi_i$ does? Since it is a representation of $A$, the domain should be $A$ and since it is a subrepresentation of $\pi$, the codomain should be $B(\mathcal{H}')$ for some subspace $\mathcal{H}' \leq \mathcal{H}$. Where is the orthogonality condition necessary?
The map, as you say, is $a\longmapsto \sum_j\pi_j(a)$. Since the $\{\mathcal H_j\}$ are pairwise orthogonal, you have no convergence issues (because $\|\sum_j T_j\|=\sup\{\|T_j\|:\ j\}$ if $T_j\in B(\mathcal H_j)$; it is essential that the subspaces are pairwise orthogonal).
If you want to emphasize the view of $\sum_j\pi_j$ as a subrepresentation, the codomain is $B(\bigoplus_j\mathcal H_j)$.
What you are doing in spirit is defining $$ \left(\sum_j\pi_j\right)(a):=(\pi_j(a))_{j\in I}. $$ The norm is the supremum, i.e., $$ \|(\pi_j(a))_{j\in I}\|=\sup\{\|\pi_j(a)\|:\ j\in I\}, $$ so everything works fine. If you had just a couple, what you are doing is $$ \left(\pi_1+\pi_2\right)(a):=\begin{bmatrix} \pi_1(a)&0\\0&\pi_2(a)\end{bmatrix}. $$