$\int_0^\lambda z de_z+\lambda p$ is a positive element of a von Neumann algebra $M$ whenever $\lambda >0$

Question

Let $M$ be a von Neumann algebra acting on a Hilbert space $\mathcal H$. Let $x$ be an positive self-adjoint operator (not necessarily bounded) such that $xu=ux$ for every unitary $u$ in $M'$. Let $x=\int_0^\infty z de_z$ be the spectral representation of $x$. Let us assume $p=1-\lim_{z\to \infty} e_z$. Since, $e_z\in M$ we have $p \in M$. Now take $x_\lambda=\int_0^\lambda z de_z+\lambda p$ for some $\lambda>0.$ I want to show that $x_\lambda$ is a positive element of $M.$
Basically $\lambda>0$, therefore we have $\displaystyle \int_0^\lambda z de_z \ge 0.$ And $p$ is a projection so $\lambda p\ge 0$. Thus $x_\lambda$ is positive. But I am unable to show that $x_\lambda \in M$. I tried to use the thing that $xu=ux$ for all unitary $u$ in $M'$ so that $x_\lambda \in M''$, but unable to apply the trick using the aforementioned argument. Please help me to solve this. Thank you for your effort.

Answer 1

Fix $u\in M'$. Let $0=a_0<a_1<\ldots<a_m=\lambda$ be a partition. Then $$ u\Big(\sum_{k=1}^ma_k\,(e_{a_{k}}-e_{a_{k-1}})\Big) =\sum_{k=1}^ma_k\,(ue_{a_{k}}-ue_{a_{k-1}}) =\sum_{k=1}^ma_k\,(e_{a_{k}}u-e_{a_{k-1}}u) =\Big(\sum_{k=1}^ma_k\,(e_{a_{k}}-e_{a_{k-1}})\Big)u. $$ As this can be done for all $u\in M'$, $$\tag1 \sum_{k=1}^ma_k\,(e_{a_{k}}-e_{a_{k-1}})\in M''=M. $$ By definition, $\displaystyle\int_0^\lambda z\,de_z$ is a norm limit of elements of the form $(1)$, so it is in $M$.