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Math Help - Projections on Neumann algebra

  1. #1
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    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!
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  2. #2
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    Quote Originally Posted by karkusha View Post
    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.
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    Thank you, Opalg. You explained that sup is in A, but why is sup projection on A?
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    Quote Originally Posted by karkusha View Post
    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).
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