σ-fragmentable Banach spaces

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§5. Some C(K) spaces that are not σ-fragmented. In [16] we remarked that the space (l∞, weak) is not σ-fragmented. Rosenthal [19] gives a number of conditions that ensure that a C(K) space contains an isomorphic copy of l∞. A Banach space is said to be injective if it is complemented in every Banach space containing it isomorphically. Rosenthal proves that every infinite dimensional injective Banach space contains a subspace isomorphic to l∞. It is known (see, for example, Day [2]) that C(K) is injective if K is extremally disconnected. Thus C(K) is not σ-fragmented when K is an infinite compact Hausdorff space that is extremally disconnected.