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By Gianni Gilardi

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21. 3 The centralizer of a Banach space In this section we shall brieﬂy discuss an algebra of operators on a Banach space X which has close relations to the M -ideal structure of X. In the same way as the M -ideals of a Banach space correspond to the closed ideals of a C ∗ -algebra, this operator algebra, called the centralizer of X, corresponds to the centre of a C ∗ -algebra (cf. 4(h)). It must be stressed that there are some diﬀerences between the real and complex case, here; cf. the Notes and Remarks.

4) By concavity of hx hx (x∗ ) ≥ λi hx (x∗i ) ≥ λi hx (x∗i ) ≥ λ1 x∗1 (x) + λ2 x∗2 (x) so that 0 ≥ λ2 x∗2 (x). In case λ2 = 0 we could have chosen x∗2 = 0, hence we may conclude x∗2 (x) ≤ 0 and thus hx (x∗2 ) = 0, hx (−x∗2 ) = −x∗2 (x). We employ the concavity of hx once again to obtain hx (−x∗ ) ≥ λi hx (−x∗i ) ≥ −λ2 x∗2 (x). Together with (4) this implies hx (x∗ ) − hx (−x∗ ) ≤ = (λ1 x∗1 + λ2 x∗2 )(x) P x∗ , x (by (1) and (2)). Since both sides of this inequality are odd functions of x∗ , the inequality is in fact an ¾ equality, and the lemma is proved.

Let {mxi | i ∈ I} be a maximal family of pairwise singular measures of this type (which exists by Zorn’s lemma) and put m(B) = i mxi (B). (Details can be found in [66, p. )] An extremally disconnected compact space admitting such a measure is called hyperstonean; equivalently, the order continuous functionals on C(Ω) (“normal measures”) separate the points of C(Ω). g. ) It is now possible to represent X as a Banach space of “integrable sections” in a product ω∈Ω Xω such that x = Ω x(ω) Xω dm(ω).