Projections on Neumann algebra

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April 21st 2011, 02:10 PM
karkusha

Projections on Neumann algebra

Hello, could you please help me to solve this problem:
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!
April 22nd 2011, 11:52 AM
Opalg

Quote:

Originally Posted by karkusha

Hello, could you please help me to solve this problem:
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.
April 22nd 2011, 12:29 PM
karkusha

Thank you, Opalg. You explained that sup is in $A$ , but why is sup projection on $A$ ?
April 22nd 2011, 12:43 PM
Opalg

Quote:

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).