Geodesic
Uncomplemented uniform algebras
Matematičeskie zametki, Tome 18 (1975) no. 1, pp. 91-96

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Let $A$ be a closed subalgebra of the algebra of all complex-valued continuous functions on a compact space $X$, and suppose $A$ contains the constant functions and separates points of $X$; let $I$ be a closed ideal of $A$ such that for some linear multiplicative functional $\varphi$ on $A$ we have the relation $0\|\varphi|_I\|1$ (for the existence of such an ideal it is sufficient that in the maximal ideal space of the algebra $A$ there exists a Gleason part consisting of at least two points). Then the Banach space $A^{**}$ is not injective [in particular, $A$ is not a complemented subspace of $C(X$)]. 
@article{MZM_1975_18_1_a11,
     author = {S. V. Kislyakov},
     title = {Uncomplemented uniform algebras},
     journal = {Matemati\v{c}eskie zametki},
     pages = {91--96},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_18_1_a11/}
}
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S. V. Kislyakov. Uncomplemented uniform algebras. Matematičeskie zametki, Tome 18 (1975) no. 1, pp. 91-96. http://geodesic.mathdoc.fr/item/MZM_1975_18_1_a11/