The operation of taking the $n$th tensor power $\mathscr L^{\otimes n}$ of a complex line bundle $\mathscr L$ over a space induces a map $\psi^n\colon\mathbb{CP}^\infty\rightarrow\mathbb{CP}^\infty$ by Yoneda. [Under the splitting principle these give the Adams operations on complex K-theory.]
I'm interested in computing the induced map on homology $H_*(\mathbb{CP}^\infty)\rightarrow H_*(\mathbb{CP}^\infty)$. What I know is that taking the infinite projective space $\mathbb C[x]/\mathbb C^\times$ as a model for $\mathbb{CP}^\infty$, the map $\mathbb{CP}^\infty\times\mathbb{CP}^\infty\rightarrow\mathbb{CP}^\infty$ representing the tensor product of line bundles is given by polynomial multiplication.
A generator in degree 2 of $H_*(\mathbb{CP}^\infty)$ is represented by the submanifold $\mathbb{CP}^1\subset\mathbb{CP}^\infty$. I would think its possible to compute its image by looking at the degree of composition $\mathbb{CP}^1\rightarrow(\mathbb{CP}^1)^n\rightarrow\mathbb{CP}^\infty\rightarrow\mathbb{CP}^1$, where the last map collapses the higher cells. Using the above description of $\mathbb{CP}^\infty$ as $\mathbb C[x]/\mathbb C^\times$, this composition is given by taking a linear polynomial $p(x)=ax+b$ to $p^n$. But I don't know what the last map collapsing the higher cells does.
Whoops, I got confused. It is less confusing to me to instead compute the induced map on cohomology, then dualize. Recall that the cohomology $H^{\bullet}(\mathbb{CP}^{\infty})$ is the polynomial algebra on a generator $c_1 \in H^2(\mathbb{CP}^{\infty})$, the universal first Chern class. The first Chern class satisfies $c_1(L_1 \otimes L_2) = c_1(L_1) + c_1(L_2)$, from which it follows pretty directly that it satisfies $c_1(L^{\otimes n}) = n c_1$ where $L$ is the universal line bundle.
This gives that $L \mapsto L^{\otimes n}$ induces the map $c_1^k \mapsto (nc_1)^k = n^k c_1^k$ on cohomology, so the map on $H^{2k}$ is multiplication by $n^k$, and this dualizes to multiplication by $n^k$ on homology.