Subalgebra of $L_\infty$

Question

Let $(\Omega,\mathcal A,\mathbb P)$ be a probability space. Let $\mathcal K$ be a unital $*$-subalgebra of $L_\infty(\Omega,\mathcal A,\mathbb P).$ Then by genearl theory of von neumann algebras we know that $\mathcal K$ is of the form $L_\infty(\Omega^\prime,\mathcal A^\prime,\mathbb P^\prime).$ Can we also take $(\Omega^\prime,\mathcal A^\prime,\mathbb P^\prime)\subseteq(\Omega,\mathcal A,\mathbb P)$?