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A commuting pair of isometries

Such a pair $(X,Y)$ is constructed as follows. Consider a Hilbert space $M$ with an orthonormal basis $\{e_n:n\in\mathbb Z\}$ and the bilateral shift $U$ on $M$ such that $Ue_n=e_{n+1}$. Denote by $S$...
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3 votes
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Spectrum of $(Jx)_n =i((2n+1)x_{n+1}-(2n-1)x_{n-1})$ on $\ell^2(\mathbb{Z})$

Under the Fourier series isomorphism $\ell^2(\mathbb{Z}) \cong L^2(-\pi,\pi)$, $u(t) = \sum_{n\in\mathbb{Z}} x_n e^{int}$, the operator becomes $$\begin{aligned} (Ju)(t) &= 4i\sin(t) u'(t) + 2i\...
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Are the ideals in two $C^*$-algebras the same?

Counterexample. Let $H = L^2(\mathbb{T}) \oplus l^2(\mathbb{N})$ and define $V_1 = M_{e^{2\pi it}} \oplus S$ and $V_2 = -I_1 \oplus I_2$. Here $M_{e^{2\pi it}}$ is a multiplication operator, $S$ is ...
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