Homology localizations after applying some right adjoint functors (Q1066515)

Each homology theory \(E_*\) determines an E*-localization in the homotopy category of CW-complexes or of CW-spectra. \textit{A. K. Bousfield} [Proc. Lond. Math. Soc., III. Ser. 44, 291-331 (1982; Zbl 0511.55007)] studied the behavior of E*-localization after application of the 0-th space functor \(\Omega^{\infty}\) and showed that the \(E_*\)-localization of an infinite loop space \(\Omega^{\infty} Y\) is still an infinite loop space. For a right adjoint functor T we introduce \(T^*E^*\)- and \((E_*,T)\)-localizations. Following our notation Bousfield's result asserts that there exists an \((E_*,\Omega^{\infty})\)-localization in the homotopy category of (- 1)-connected CW-spectra. This is equivalent to saying that there exists an \(\Omega^{\infty^*}E_*\)-localization and moreover ``\(\Omega^{\infty} \Sigma^{\infty} f:\) \(\Omega^{\infty} \Sigma^{\infty} X\to \Omega^{\infty} \Sigma^{\infty} Y\) is an \(E_*\)-equivalence if f is one''. \textit{N. J. Kuhn} [Trans. Am. Math. Soc. 283, 303-313 (1984; Zbl 0544.55016)] gave a simple proof of the latter condition independently. In this note, by use of only this latter condition we give a direct proof of the existence theorem of \((E_*,\Omega^{\infty})\)-localization, without the knowledge of very special \(\Gamma\)-spaces.