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For every commutative C$^*$-algebra the spectral radius is equal to the norm. My question is:

For which C$^*$-algebras $\mathcal A$ does there exist a constant $C>0$ such that $$C\|a\| \leq r(a), \ \ \forall a\in \mathcal A?$$

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    $\begingroup$ The answer (which is A must be commutative) is basically contained in this question mathoverflow.net/questions/125945/… : If A is non-commutative it has a non-zero nilpotent element which has spectral radius 0 $\endgroup$ Jul 26, 2016 at 16:31
  • $\begingroup$ Delightful! Thank you @CalebEckhardt. $\endgroup$ Jul 26, 2016 at 16:34

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