Let $M$ be a type $II\_{1}$ factor, Let $B$ is an infinite dimensional nonabelian subalgebra. Is it true that $B$ always type $II\_{1}$ ?



Subalgebras of $II_{1}$ factor
If $M$ is type $II\_1$ then it has a tracial state, and hence so does $B$ by restriction. So if $B$ is an infinite dimensional subfactor it must be type $II\_1$, by inspection of factor types. If $B$ is merely a subalgebra then it could have a type $I$ part, e.g. $\mathbb{C}p + qMq$ where $p$ and $q$ are nonzero projections with $p+q=1$.


