Primitive and decomposable elements in homology of ωςℂP^∞

For each positive integer n n, we let φ n: ς C P ∞ → ς C P ∞ be the self-maps of the suspension of the infinite complex projective space, or the localization of this space at a set of primes which may be an empty set. Furthermore, let [ φ m, φ n ]: ς C P ∞ → ς C P ∞ be a commutator of self-maps φ m for any positive integers m m and n n. In the current study, we show that the image of the homomorphism [ φ m, φ n ] in homology induced by the adjoint [ φ m, φ n ]: C P ∞ → ω ς C P ∞ of the commutator [ φ m, φ n ] is both primitive and decomposable. As a further support of the above statement, we provide an example.