Monic projections in finite von Neumann algebra

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The observation and heart of the proof of existence of trace lies on the fact in finite vN algebra any projection is orthogonal sum of monic projections, can somebody reveal me the idea and motivation for defining the object "monic projections" to generalize the trace for infinte dimensional (finite) vN algebras?? From where the idea come from? I am really curious to know these.
Thanks




operator-algebras c-star-algebras von-neumann-algebras




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    up vote
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    down vote

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    The observation and heart of the proof of existence of trace lies on the fact in finite vN algebra any projection is orthogonal sum of monic projections, can somebody reveal me the idea and motivation for defining the object "monic projections" to generalize the trace for infinte dimensional (finite) vN algebras?? From where the idea come from? I am really curious to know these.
    Thanks




    operator-algebras c-star-algebras von-neumann-algebras




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      The observation and heart of the proof of existence of trace lies on the fact in finite vN algebra any projection is orthogonal sum of monic projections, can somebody reveal me the idea and motivation for defining the object "monic projections" to generalize the trace for infinte dimensional (finite) vN algebras?? From where the idea come from? I am really curious to know these.
      Thanks




      operator-algebras c-star-algebras von-neumann-algebras




      share|cite|improve this question











      The observation and heart of the proof of existence of trace lies on the fact in finite vN algebra any projection is orthogonal sum of monic projections, can somebody reveal me the idea and motivation for defining the object "monic projections" to generalize the trace for infinte dimensional (finite) vN algebras?? From where the idea come from? I am really curious to know these.
      Thanks





      operator-algebras c-star-algebras von-neumann-algebras





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      asked Aug 1 at 19:10









      mathlover

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          The idea is exactly modeled from what happens in finite dimension. If $$ A=bigoplus_k=1^m M_n_k(mathbb C),$$ then you have minimal central projections $p_1,ldots,p_m$ corresponding to each block. The $(n_1,p_1)$-monic projections (I'm using Blackadar's notation, today was the first time in my life I encountered the term "monic projection") are the rank-one projections from the first block. Because $p_1=e_11+cdots+e_n_1,n_1$, and any two rank-one projections in the first block are equivalent.






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          • The term "monic" is defined in Kadison, Sir.
            – mathlover
            Aug 2 at 6:50






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            Fair enough. $ $
            – Martin Argerami
            Aug 2 at 16:30










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          The idea is exactly modeled from what happens in finite dimension. If $$ A=bigoplus_k=1^m M_n_k(mathbb C),$$ then you have minimal central projections $p_1,ldots,p_m$ corresponding to each block. The $(n_1,p_1)$-monic projections (I'm using Blackadar's notation, today was the first time in my life I encountered the term "monic projection") are the rank-one projections from the first block. Because $p_1=e_11+cdots+e_n_1,n_1$, and any two rank-one projections in the first block are equivalent.






          share|cite|improve this answer





















          • The term "monic" is defined in Kadison, Sir.
            – mathlover
            Aug 2 at 6:50






          • 1




            Fair enough. $ $
            – Martin Argerami
            Aug 2 at 16:30














          up vote
          1
          down vote



          accepted










          The idea is exactly modeled from what happens in finite dimension. If $$ A=bigoplus_k=1^m M_n_k(mathbb C),$$ then you have minimal central projections $p_1,ldots,p_m$ corresponding to each block. The $(n_1,p_1)$-monic projections (I'm using Blackadar's notation, today was the first time in my life I encountered the term "monic projection") are the rank-one projections from the first block. Because $p_1=e_11+cdots+e_n_1,n_1$, and any two rank-one projections in the first block are equivalent.






          share|cite|improve this answer





















          • The term "monic" is defined in Kadison, Sir.
            – mathlover
            Aug 2 at 6:50






          • 1




            Fair enough. $ $
            – Martin Argerami
            Aug 2 at 16:30












          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          The idea is exactly modeled from what happens in finite dimension. If $$ A=bigoplus_k=1^m M_n_k(mathbb C),$$ then you have minimal central projections $p_1,ldots,p_m$ corresponding to each block. The $(n_1,p_1)$-monic projections (I'm using Blackadar's notation, today was the first time in my life I encountered the term "monic projection") are the rank-one projections from the first block. Because $p_1=e_11+cdots+e_n_1,n_1$, and any two rank-one projections in the first block are equivalent.






          share|cite|improve this answer













          The idea is exactly modeled from what happens in finite dimension. If $$ A=bigoplus_k=1^m M_n_k(mathbb C),$$ then you have minimal central projections $p_1,ldots,p_m$ corresponding to each block. The $(n_1,p_1)$-monic projections (I'm using Blackadar's notation, today was the first time in my life I encountered the term "monic projection") are the rank-one projections from the first block. Because $p_1=e_11+cdots+e_n_1,n_1$, and any two rank-one projections in the first block are equivalent.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Aug 1 at 20:16









          Martin Argerami

          115k1071164




          115k1071164











          • The term "monic" is defined in Kadison, Sir.
            – mathlover
            Aug 2 at 6:50






          • 1




            Fair enough. $ $
            – Martin Argerami
            Aug 2 at 16:30
















          • The term "monic" is defined in Kadison, Sir.
            – mathlover
            Aug 2 at 6:50






          • 1




            Fair enough. $ $
            – Martin Argerami
            Aug 2 at 16:30















          The term "monic" is defined in Kadison, Sir.
          – mathlover
          Aug 2 at 6:50




          The term "monic" is defined in Kadison, Sir.
          – mathlover
          Aug 2 at 6:50




          1




          1




          Fair enough. $ $
          – Martin Argerami
          Aug 2 at 16:30




          Fair enough. $ $
          – Martin Argerami
          Aug 2 at 16:30












           

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