# Thread: Projections on Neumann algebra

1. ## Projections on Neumann algebra

let A is a von Neumann algebra, and ( $p_i$) - projections on A, $i\in I$. Then inf $p_i$, sup $p_i$ - projections on A.
Thank you!

2. Originally Posted by karkusha
let A is a von Neumann algebra, and ( $p_i$) - projections on A, $i\in I$. Then inf $p_i$, sup $p_i$ - projections on A.
If $p$, $q$ are two projections in $A$, then $p\vee q$ is the range projection of $p+q$, and is therefore in $A.$ By induction, the sup of any finite family of projections in $A$ is also in $A.$

Given an infinite family, the directed net of sups of finite subfamilies (ordered by inclusion) is increasing, and bounded above by the identity. Therefore it converges strongly to a limit which is a projection and is therefore the sup of the whole family.

The result for infs follows by taking orthogonal complements.

3. Thank you, Opalg. You explained that sup is in $A$, but why is sup projection on $A$?

4. Originally Posted by karkusha
Thank you, Opalg. You explained that sup is in $A$, but why is sup projection on $A$?
The sup of an increasing directed net of projections is equal to the sup projection (Kadison and Ringrose, Proposition 2.5.6).