Is the entangibility of density operators relied on what component spaces are being specified?
More precisely, let $H$ be a Hilbert space, $\rho$ be a density operator on $H$. Suppose we were not given any information about $H$. i.e., it is possible that $H=H_A\otimes H_B$, or $H=H_C\otimes H_D$, and perhaps that $\dim H_A\neq\dim H_C$ and $\dim H_B\neq\dim H_D$. If $\rho$ is, say being entangled with respect to $H_A$ and $H_B$, is $\rho$ also necessarily entangled with respect to $H_C$ and $H_D$?