Definition 38.5.10 (Direct Sum).label Let $A$ be a $C^{*}$-algebra and $\bracsn{(H_i, \pi_i)}_{i \in I}$ be representations of $A$, then the representation
\[\pi: A \to B([l^{2}(I); H_{i}]) \quad \pi(x)(\xi)_{i} = \pi_{i}(x)(\xi_{i})\]
is the direct sum of $\bracsn{(H_i, \pi_i)}_{i \in I}$, denoted $\bigoplus_{i \in I}\pi_{i}$.
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