Let $A$ be a unital C\*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?



Density of normal elements in a C*- algebra
The normal elements of any $C^\*$-algebra $A$ are norm-closed in $A$. Indeed, if $\{a\_n\}\_n$ is a sequence of normal elements with $a= \lim\_n a\_n$, then by joint continuity of multiplication and involution,
$$aa^\* = \lim\_n a\_n a\_n^\* = \lim\_n a\_n^\* a\_n = a^\*a.$$


Hence, your question is equivalent with: when is every element of a $C^\*$-algebra normal?


By the comment of @Jochen Glueck below, this is only the case when $A$ is commutative.


