I had the following question when I am learning von Neumann algebras: 
Let p be a finite projection in a finite von Neumann algebra $M$, let $p>p\_1>p\_2>\cdots$ be a decreasing sequence of projections such that $p\_i\neq p\_{i+1}$. Is such a sequence necessarily of finite length? 
Any help or recommendations on books/papers are greatly appreciated.



Finite projection in Von Neumann algebra
The answer to your question is **no**. Abelian von Neumann algebras are finite, but it is easy to find examples of such algebras with infinite decreasing sequences of projections. For instance, one may take $M = L^\infty(\mathbb{R},\lambda)$ where $\lambda$ is the Lebesgue measure, and let $p\_i \in M$ be the element corresponding to the characteristic function of $[0,\frac{1}{i}]$.


**Update:** Now that you've seen some counterexamples, this might be a good follow-up "homework" problem for you (depending on how much you know about [measure theory](http://en.wikipedia.org/wiki/Maharam%27s_theorem) and [abelian von Neumann algebras](http://en.wikipedia.org/wiki/Abelian_von_Neumann_algebra#Classification)): prove that a (finite) von Neumann algebra $M$ satisfies your descending chain condition on projections if and only if it is a finite dimensional algebra. 


(Hint: think about a type decomposition.)


