Is a projection infinite iff its range is infinite-dimensional?

Question

A projection is said to be infinite if it is Murray-vN equivalent to some subprojection. In other words, there exists $q,a\in A$ such that $q\leq p$, $p=aa^*$ and $q=a^*a$.

Is this condition equivalent to the projection having infinite-dimensional range (or perhaps a finite-dimensional kernel)?

If not, I would appreciate a counter-example.

Answer 1

Certainly not. If $\mathcal H$ is an infinite-dimensional Hilbert space, and $A=\mathbb C\cdot1_{\mathcal H}$, then $1_{\mathcal H}$ is a finite projection in $A$, but has infinite-dimensional range.