Let $\mathfrak{H}$ be a Hilbert space and let $\mathcal{E}(\mathfrak{H})$ be the set of all operators $T\in B(\mathfrak{H})$ such that $0\leq T\leq 1$ (these operators are also called effects on $\mathfrak{H}$ and $\mathcal{E}(\mathfrak{H})$ is also called effect algebra). Now let $\alpha:\mathcal{E}(\mathfrak{H})\rightarrow\mathcal{E}(\mathfrak{H})$ be a bijection such that the following two properties hold:
- $\alpha(tA+(1-t)B)=t\alpha(A)+(1-t)\alpha(B)$ for all $0<t<1$ and all $A,B\in\mathcal{E}(\mathfrak{H})$ (hence $\alpha$ preserves the convex structure of the effect algebra).
- $A\leq B \Longrightarrow \alpha(A)\leq\alpha(B)$ (hence $\alpha$ respects the ordening)
The claim is now the following: For every $\alpha$ with this properties, there exists an operator $U:\mathfrak{H}\rightarrow\mathfrak{H}$ which is either unitary or anti-unitary such that $\alpha(A)=UAU^*$ for each $A\in\mathcal{E}(\mathfrak{H})$.
This result seems like the theorems by Gleason and Ludwig (more Ludwig's Theorem), but can anyone can help me with this because I don't know how to start or where I have to search for this.
Thank you very much.