Assume that $GL(\mathcal{H})$ is the group of invertible linear transformations of Hilbert space $\mathcal{H}$ and $U(\mathcal{H})$ is the group of unitary operators on $\mathcal{H}$. We know that $\mathbb{C}^*$ and $U(1)$ act respectively on $GL(\mathcal{H})$ and $U(\mathcal{H})$.
Is $GL(\mathcal{H})/\mathbb{C}^* \cong U(\mathcal{H})/U(1)$?