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nLab > Latest Changes: von Neumann regular ring

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2024

    a stub entry, for the moment just to make the link work

    v1, current

    • CommentRowNumber2.
    • CommentAuthorJohn Baez
    • CommentTime2 days ago

    I included a bunch of equivalent definitions of ’von Neumann regular ring’, and a proof of what’s essentially Lemma 5 from von Neumann’s original paper. This clarifies the funny-looking equation a=axaa = a x a.

    My lemma and proof are not correctly formatted. Maybe someone could do that? I always have to look up how to do it.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorJ-B Vienney
    • CommentTime2 days ago
    • (edited 2 days ago)

    I don’t know if it’s the best way but I just wrote \begin{lemma} \end{lemma} \begin{proof} \end{proof} and it works fine.

    • CommentRowNumber4.
    • CommentAuthorJohn Baez
    • CommentTime2 days ago

    I showed all the first 5 definitions of von Neumann regular ring are equivalent.

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorJohn Baez
    • CommentTime2 days ago

    Added an “idea” section sketching some motivation from quantum foundations.

    diff, v5, current

  6. started examples section

    Anonymouse

    diff, v7, current

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTime1 day ago

    More examples

    diff, v8, current

    • CommentRowNumber8.
    • CommentAuthorJohn Baez
    • CommentTime6 hours ago

    Deleted my claim that von Neumann algebras are von Neumann regular rings. This now seems false to me, since I think L ∞(ℝ)L^\infty(\mathbb{R}) has principal ideals that aren’t generated by idempotents (i.e. characteristic functions). For example, the ideal generated by a continuous ’bump function’.

    diff, v10, current

1 to 8 of 8

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