# A question about projections in a von Neumann algebra

Let $\cal{R}$ be a von Neumann algebra $P\in\cal{R}$ a projection, $G$ a subprojection of $P$. Then the central carrier of $G+(I-C_G)E$ is $C_E$.

Note: The central carrier of a projection $P$ is defined as the intersection of all central projections $Q$ such that $QP=P$, and is denoted by $C_P$.

When I read Kadison's Fundamentals of the theory of operator algebras, I met the above statement which I cannot prove. It is clear that the central carrier of $G+(I-C_G)E$ is a subprojection of $C_E$, but I do not know how to prove that $C_E$ is a subprojection of the central carrier of $G+(I-C_G)E$.

Thanks a lot to the anyone who can give me a hint.

I suppose $P=E$ in your notation. Only need to check $TE=E$ where $T$ is the central carrier of $G+(I-C_G)E$.
Toward this, first use definition of $T$' we have $T(G+(I-C_G)E)=G+(I-C_G)E$, multiply both side by $G$, we get $TG=G$, so $C_G\leq T$, then use this relation and $TG=G$ to simplify $T(G+(I-C_G)E)=G+(I-C_G)E$, you get what we want, i.e., $TE=E$