Multiplicative properties of positive maps

Abstract

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

8 pages