Let [tex]X[/tex] be a [tex]C^*[/tex]-algebra. I know that if [tex]x\in X[/tex] is self-adjoint, then its spectrum is real, [tex]\sigma(x)\subset\mathbb{R}[/tex]. I haven't seen a claim about the converse, but it seems difficult to come up with a counter example for it. My question is, that is it possible, that some [tex]x\in X[/tex] has a real spectrum, but still [tex]x^*\neq x[/tex]?(adsbygoogle = window.adsbygoogle || []).push({});

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# Real spectrum, not self-adjoint

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