finite projection in a semifinite von Neumann algebra

Question

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?

Answer 1

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 $$ \rho(p)=\rho\bigg(p-\sum_kq_k\bigg)+\sum_k\rho(q_k)\leq(n+1)\rho(q)<\infty. $$