# finite projection in a semifinite von Neumann algebra

Let $$M$$ be a semifinite von Neumann algebra equipped with a semifinite normal faithful trace $$\rho$$. Let $$p\in M$$ be a nonzero projection in $$M$$.

If $$\rho(p)=\infty$$, can we deduce that $$p$$ is not a finite projection?

If $$ρ(p)=∞$$, can we deduce that $$p$$ is not a finite projection?
No, this can fail. For instance let $$M=\ell^\infty(\mathbb N)$$ with the fns trace $$\rho(x)=\sum_n\frac1n\,x_n.$$ We have $$\rho(1)=\infty$$, while every projection is finite since the algebra is abelian.
On a factor, on the other hand, the assertion is true. If $$p$$ is finite, then the semifinitness guarantees that there exists $$q\leq p$$ with $$\rho(q)<\infty$$. Let $$\{q_1,\ldots,q_n\}$$ be a maximal family of pairwise orthogonal projections such that $$q_k\sim q$$ and $$q_k\leq p$$ for all $$k$$. The family is necessarily finite for otherwise $$p$$ would be infinite. We have that $$p-\sum_kq_k\prec q$$. Then $$\rho(p)=\rho\bigg(p-\sum_kq_k\bigg)+\sum_k\rho(q_k)\leq(n+1)\rho(q)<\infty.$$