Let $(M,\tau)$ be a tracial von Neumann algebra, i.e.

• a unital subalgebra $M=M''\subset \mathbb{B}(H)$;
• a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ implies $x=0$; moreover $\tau(q)\leq 1$ for any projection $q\in M$).

Consider a projection $p\in M$, then its central support (denoted by $z(p)$) is defined as the smallest projection in the centre $Z(M)=M\cap M'$ greater than $p$. It is known that

• $z(p)=\vee_{u\in U(M)} upu^*$, where $U(M)$ are the unitaries of $M$ (i.e. elements $u$ such that $uu^*=u^*u=1$).

I read that under the above hypotheses one can prove that there exists a finite number of projections $\{p_i\}_{i=1}^n$ such that $p_i\leq p$, $p_i\in M$ and $z(p)=\sum_{i=1}^np_i$. I think that the above characterization of $z(p)$ should be useful in the proof of this result but I have not been able to make a proof. Can anyone help me understanding why the above claim is true? Thank you in advance for the help.

Let $(M,\tau)$ be a tracial von Neumann algebra, i.e.

• a unital subalgebra $M=M''\subset \mathbb{B}(H)$;
• a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ implies $x=0$; moreover $\tau(q)\leq 1$ for any projection $q\in M$).

Consider a projection $p\in M$, then its central support (denoted by $z(p)$) is defined as the smallest projection in the centre $Z(M)=M\cap M'$ greater than $p$. It is known that

• $z(p)=\vee_{u\in U(M)} upu^*$, where $U(M)$ are the unitaries of $M$ (i.e. elements $u$ such that $uu^*=u^*u=1$).

I read that under the above hypotheses one can prove that there exists a finite number of projections $\{p_i\}_{i=1}^n$ such that $p_i\preceq p$, $p_i\in M$ and $z(p)=\sum_{i=1}^np_i$. I think that the above characterization of $z(p)$ should be useful in the proof of this result but I have not been able to make a proof. Can anyone help me understanding why the above claim is true? Thank you in advance for the help.