Multiplicative properties of positive maps

Abstract

Let $ϕ$ be a positive unital normal map of a von Neumann algebra $M$ into itself, and assume there is a family of normal $ϕ$-invariant states which is faithful on the von Neumann algebra generated by the image of $ϕ$. It is shown that there exists a largest Jordan subalgebra $C_ϕ$ of $M$ such that the restriction of $ϕ$ to $C_ϕ$ is a Jordan automorphhism, and each weak limit point of $(ϕ^n (a))$ for $a\in M$ belongs to $C_ϕ$.