Definition 45.2.5 (Amplification).label Let $(M, \tau)$ be a $\vnII_{1}$ factor, $t > 0$, $n \in \natp$ with $n \ge t$, and $p \in \text{Proj}(M_{n}(M))$ with $(\text{Tr}_{n} \otimes \tau)p = t$, then the von Neumann algebra $M^{t} := pM_{n}(M)p \subset M_{n}(M)$ is an amplification of $M$.

The spatial isomorphism class of $M^{t}$ is independent of the choice of $n \ge t$ and $p \in \text{Proj}(M_{n}(M))$ with $(\text{Tr}_{n} \otimes \tau)p = t$.