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Are the ideals in two $C^*$-algebras the same?

Let $V_{1}, V_{2}$ be the commuting isometries. By Wold decomposition theorem, we know that $V_{i}$ admits decomposition $$V_i \cong V^s_{i}\oplus V^{u}_{i},$$ where $V^{s}_{i}$ is the shift and $V^{u …
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