For a finite CW-complex $X$, the K-theory group $K^{-1}(X)$ is isomorphic to the group of homotopy classes of maps $[X, U(\infty)]$. The group of isomorphism classes of line bundles on $X$, which I denote by $\text{Pic}(X)$, is isomorphic to $[X, \mathbb{CP}^\infty]$, and it acts on $K^{-1}(X)$ by tensor product.

My question: does anybody know how to write down explicitly the action of $\mathbb{CP}^\infty$ on $U(\infty)$ which induces the action of $\text{Pic}(X)$ on $K^{-1}(X)$?

Note: Of course the action of $\mathbb{CP}^\infty$ on $U(\infty)$ should be compatible with the H-space structure of $\mathbb{CP}^\infty$, which induces the tensor product of line bundles. For this H-space structure see this post, which inspires my question.