[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that we have an isomorphism

[[math]] C(\widetilde{S}_2^+)\simeq\left\{(x,y)\in C^*(D_\infty)\oplus C^*(D_\infty)\Big|\varepsilon(x)=\varepsilon(y)\right\} [[/math]]

where [math]\varepsilon:C^*(D_\infty)\to\mathbb C1[/math] the usual counit map.