Hello, could you please help me to solve this problem:
let A is a von Neumann algebra, and ( ) - projections on A, . Then inf , sup - projections on 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.