Definition 38.5.9 (Intertwining Operator).label Let $A$ be a $C^{*}$-algebra, $(H_{1}, \pi_{1})$ and $(H_{2}, \pi_{2})$ be representations of $A$, and $T \in L(H_{1}; H_{2})$, then $T$ is an intertwining operator for $\pi_{1}$ and $\pi_{2}$ if the following diagram commutes
\[\xymatrix{ H_1 \ar@{->}[d]_{\pi_1(x)} \ar@{->}[r]^{T} & H_2 \ar@{->}[d]^{\pi_2(x)} \\ H_1 \ar@{->}[r]_{T} & H_2 }\]
for all $x \in A$. The set $\mathcal{C}(\pi_{1}, \pi_{2})$ is the space of intertwining operators for $\pi_{1}$ and $\pi_{2}$.
In particular, $\mathcal{C}(\pi_{1}, \pi_{1}) = \pi(A)'$, and hence is a von Neumann algebra.
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