Let $N$ be a Type $II_\infty$ factor, and let $\operatorname{Tr}:N_+\to[0,\infty]$ be a faithful normal semifinite trace whose finite part has the usual linear tracial extension on each finite-trace hereditary subalgebra. If $p\in N$ is a nonzero projection such that $\operatorname{Tr}(p)<\infty$, then the corner von Neumann algebra $pNp$ is a Type $II_1$ factor with identity $p$. Moreover, the formula

\begin{align*} \tau_p(x)=\frac{\operatorname{Tr}(x)}{\operatorname{Tr}(p)} \end{align*}

for $x\in(pNp)_+$ defines a faithful normal tracial state on the positive cone of $pNp$, and this positive functional extends uniquely by linearity to a faithful normal tracial state $\tau_p:pNp\to\mathbb C$.