matrix algebras of von Neumann algebra factor of type $II1$

Question

Suppose $M$ is a von Neumann algebra factor of type $II_1$ ,Is $\Bbb M_n(M),n\in \Bbb N$ also a von Neumann algebra factor of type $II_1$ ?

Answer 1

Yes this is true. $\mathbb M_n(M) \cong \mathbb M_n(\mathbb C) \overline{\otimes} M$. Tensor product of factors is always a factor. $\mathbb M_n(M)$ is clearly infinite dimensional. Moreover, $\mathbb M_n(M)$ has a tracial state given by $tr_n \otimes \tau$, where $\tau$ is the trace on $M$, and $tr_n$ denotes the normalized trace on $\mathbb M_n(\mathbb C)$.

Hence $\mathbb M_n(M)$ is a $\rm II_1$ factor.