Definition 36.5.5 (Unitary Equivalence).label Let $A$ be a $C^{*}$-algebra and $(H_{1}, \pi_{1}), (H_{2}, \pi_{2})$ be representations of $A$, then $(H_{1}, \pi_{1})$ and $(H_{2}, \pi_{2})$ are unitarily equivalent if there exists an isometry $U \in L(H_{1}; H_{2})$ such that the following diagram commutes
\[\xymatrix{ H_1 \ar@{->}[r]^{U} \ar@{->}[d]_{\pi_1(x)} & H_2 \ar@{->}[d]^{\pi_2(x)} \\ H_1 & H_2 \ar@{->}[l]^{U^*} }\]
for all $x \in A$.
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