Rational circle-equivariant elliptic cohomology of CP(V)

$\def\T{\mathbb{T}}\def\CPV{\mathbb{C}P(V)}$ We prove a splitting result between the algebraic models for rational $\T^2$- and $\T$-equivariant elliptic cohomology, where $\T$ is the circle group and $\T^2$ is the $2$-torus. As an application we compute rational $\T$-equivariant elliptic cohomology of $\CPV$: the $\T$-space of complex lines for a finite dimensional complex $\T$-representation $V$. This is achieved by reducing the computation of $\T$-elliptic cohomology of $\CPV$ to the computation of $\T^2$-elliptic cohomology of certain spheres of complex representations.