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 factor 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 $p+q=1$.
Nik Weaver
- 42.8k
- 3
- 112
- 213