Definition:Direct Sum of Representations of C*-Algebra

Definition

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a $\text C^\ast$-algebra.

Let $\family {\tuple {\pi_i, \HH_i} }_{i \mathop \in I}$ be an $I$-index family of representations of $\struct {A, \ast, \norm {\, \cdot \,} }$.

Let:

$\ds \HH = \bigoplus_{i \mathop \in I} \HH_i$

be the Hilbert space direct sum of $\sequence {\struct {\HH_i, \innerprod \cdot \cdot_i} }_{i \mathop \in I}$ with norm $\norm {\, \cdot \,}$.

Let:

$\ds \pi = \bigoplus_{i \mathop \in I} \pi_i$

be the direct sum of $\family {\pi_i}_{i \mathop \in I}$.

We say that $\tuple {\pi, \HH}$ is the direct sum of $\family {\tuple {\pi_i, \HH_i} }_{i \mathop \in I}$.