Primitive and decomposable elements in homology of ΩΣℂP ∞

For each positive integer n n , we let φ n : Σ C P ∞ → Σ C P ∞ {\varphi }_{n}:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty } 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 ∞ \left[{\varphi }_{m},{\varphi }_{n}]:\Sigma {\mathbb{C}}{P}^{\infty }\to \Sigma {\mathbb{C}}{P}^{\infty } be a commutator of self-maps φ m {\varphi }_{m} and φ n {\varphi }_{n} for any positive integers m m and n n . In the current study, we show that the image of the homomorphism [ φ ˆ m , φ ˆ n ] ∗ {\left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]}_{\ast } in homology induced by the adjoint [ φ ˆ m , φ ˆ n ] : C P ∞ → Ω Σ C P ∞ \left[{\hat{\varphi }}_{m},{\hat{\varphi }}_{n}]:{\mathbb{C}}{P}^{\infty }\to \Omega \Sigma {\mathbb{C}}{P}^{\infty } of the commutator [ φ m , φ n ] \left[{\varphi }_{m},{\varphi }_{n}] is both primitive and decomposable. As a further support of the above statement, we provide an example.