On the mod p decomposition of \(Q({\mathbb{C}}P^{\infty})\) (Q1058193)

\textit{G. Segal} [Q. J. Math., Oxf. II. Ser. 24, 1-5 (1973; Zbl 0266.55009)] showed that the natural map \(\lambda\) : Q(\({\mathbb{C}}P^{\infty})\to BU\), induced by the Hopf bundle, is split. Hence \(Q({\mathbb{C}}P^{\infty})\simeq BU\times F\), where F is a space with finite homotopy groups. \textit{M. Mimura}, \textit{G. Nishida} and \textit{H. Toda} [J. Math. Soc. Japan 23, 593-624 (1971; Zbl 0217.488)] showed that \(\Sigma {\mathbb{C}}P_{(p)}\simeq \bigvee_{1\leq k\leq p-1}X_ k\), where \(Y_{(p)}\) denotes the p-localisation of Y. The author relates these two results by showing that for each \(1\leq k\leq p-1\) there is a splitting of the form \(\Omega Q(X_ k)\simeq G_ k\times F_ k\) where \(G_ k\) is the k-th factor in the p-local splitting of BU obtained by \textit{J. F. Adams} in [Proc. Conf. Seattle Res. Center Battelle Mem. Inst. 1968, 3 (Lect. Notes Math. 99), 1-138 (1969; Zbl 0193.517)]. The reviewer [Mem. Am. Math. Soc. 221 (1979; Zbl 0413.55004)] showed that Segal's splitting is a decomposition of H-spaces. The same extension is true of the author's result.