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\begin{lemma}
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\label{lemma:lcg-intertwining}
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Let $G$ be a locally compact group and $(H, \pi)$ be a unitary representation of $G$, then $\mathcal{C}(\pi, \pi)$ is the commutant of $\pi(G)'$, and hence a von Neumann algebra.
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Let $G$ be a locally compact group and $(H, \pi)$ be a unitary representation of $G$, then $\mathcal{C}(\pi, \pi) = \pi(G)'$, and hence a von Neumann algebra.
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\end{lemma}
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\begin{definition}[Invariant Subspace]
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