Hello, could you please help me to solve this problem:
let A is a von Neumann algebra, and ($\displaystyle p_i $) - projections on A, $\displaystyle i\in I$. Then inf $\displaystyle p_i$, sup $\displaystyle p_i$ - projections on A.
Thank you!
Hello, could you please help me to solve this problem:
let A is a von Neumann algebra, and ($\displaystyle p_i $) - projections on A, $\displaystyle i\in I$. Then inf $\displaystyle p_i$, sup $\displaystyle p_i$ - projections on A.
Thank you!
If $\displaystyle p$, $\displaystyle q$ are two projections in $\displaystyle A$, then $\displaystyle p\vee q$ is the range projection of $\displaystyle p+q$, and is therefore in $\displaystyle A.$ By induction, the sup of any finite family of projections in $\displaystyle A$ is also in $\displaystyle A.$
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.
The sup of an increasing directed net of projections is equal to the sup projection (Kadison and Ringrose, Proposition 2.5.6).