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
2026-03-25 19:03:53.1774465433
Monic projections in finite von Neumann algebra
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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.