In case of unital $C^*$ algebras $A$ and $B$ we have that any $^*$-isomorphism preserves norm. Now my question is that what if we take non-unital $C^*$ algebras does $^*$-isomorphism preserves norm i.e $$ \phi:C \rightarrow D$$ *-isomorphism . $C^*$ algebrasDoes $\||\phi(c)\||=\||c\||$ for $c\in C.$ In case of unital we use the spectral radius to prove the result.
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Can't you just extend $\phi$ to the unitalisations of $C$ and $D$ and then apply your result? – WoolierThanThou Aug 02 '20 at 07:25