Multiplicative properties of positive maps

Let $\phi$ be a positive unital normal map of a von Neumann algebra $M$ into itself. It is shown that with some faithfulness assumptions on $\phi$ there exists a largest Jordan subalgebra $C_{\phi}$ of $M$ such that the restriction of $\phi$ to $C_{\phi}$ is a Jordan automorphism and each weak limit point of $(\phi^n (a))$ for $a\in M$ belongs to $C_{\phi}$.