I have a question about commutative direct summands of $C$\*-algebras.


Let $A$ be a $C$\*-algebra (with unit) and suppose that its bidual $A^{\*\*}$ has a commutative direct summand, that is, $A^{\*\*}=B\oplus C$ such that $B$ is non-zero and commutative. Does this force $A$ to have a non-zero commutative direct summand as well?


Thank you very much in advance for your feedback!



Commutative direct summands of C*-algebras
Let $A$ be the continuous functions $f$ from $[0,2]$ into $M\_2(\mathbb{C})$ such that $f(t)$ is diagonal for $0 \leq t \leq 1$. Then $A$ has no commutative direct summand, but the atomic part of its bidual should be equal to the bounded functions $f$ from $[0,2]$ into $M\_2(\mathbb{C})$ such that $f(t)$ is diagonal for $0 \leq t \leq 1$, which contains plenty of commutative direct summands.


