We can classify von Neumann algebras into semi-finite von Neumann algebras and non-semi-finite ones.
I know the fact there are type I,II,III von Neumann algebras.
Do there exist relationships between semi-finiteness and three types?
2026-03-30 12:58:59.1774875539
classifiction of von Neumann algebras
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Yes. Types I and II are semi-finite, and III are not. Semi-finiteness is equivalent to having a tracial weight, which can never happen in type III.
Never forget that a von Neumann algebra is not necessarily of one of the three types. Rather, it has central summands (or even, central integrands if the algebra is a direct integral) of some of the three types.