I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does not have a non-zero commutative algebra direct summand:

Consider the $C^*$-algebra $$ A:=\bigl\{f\in C([0,2];\mathrm{M}_2(\mathbb{C}))\mid \ f \mbox{ is diagonal on } [0,1]\bigr\}. $$ Then $A$ does not have a non-zero commutative algebra direct summand, but the atomic part of the bidual should be $$ A^{**}_\mathrm{atomic}=\bigl\{f\in\ell^\infty([0,2];\mathrm{M}_2(\mathbb{C}))\mid f \mbox{ is diagonal on } [0,1]\bigr\} $$ which has plenty of central atoms. See this question.

It is true that the bidual of a $C^*$-algebra $A$ has no central atoms if and only if the state space $S(A)$ of $A$ has no singleton split faces. This is Corollary 5.33 and Corollary 5.34 in *Geometry of State Spaces of Operator Algebras* by Alfsen and Shultz. This characterises these algebras in terms of the state space, but knowing a condition on $A$ would be very interesting.

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