Let $M$ be a type $II_1$ factor,is it true that two projections in $M$ are equivalent iff they have the same trace?
what is the precise definition of $II_1$ factor?Is it an infinite dimensional factor with a unique normal tracial state?
2026-03-26 20:16:59.1774556219
a von Neumann algebra factor of type $II_1$
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The most common definition is that a II$_1$-factor is a finite factor (that is, the identity is finite as a projection) with no minimal projections. This implies the existence of a unique faithful trace.
An infinite-dimensional factor with a unique faithful (this is important) trace is II$_1$ (the normality follows). This is easy to see, since the existence of the trace precludes types I$_\infty$, II$_\infty$ and III, and infinite-dimensionality precludes types I$_n$.
And, as you say, on a II$_1$ factor if two projections have the same trace, then they are equivalent. This follows directly from the fact that due to comparison (and faithfulness of the trace), non-equivalent projections have different trace.