Let $(A,\|.\|)$ be a complex unital Banach algebra. Let $r(a)$ the spectral radius of an element in $A$. Suppose the spectral radius is a norm on $A$, then does that imply that $\|a\|\leq k r(a)$ for every $a\in A$ and some $K>0$? This is with reference to the converse of Corollary 7 here.
2026-03-25 12:20:55.1774441255
The spectral radius is a norm
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It doesn't.
Corollary 5 talks about "another norm" $\|\cdot\|_1$. That other norm is now $\|\cdot\|_\sigma$, with $k=1$.