Equivalence of projections in a von Neumann agebra Can anyone reveal me what was von Neumann and Murray's original idea behind the definition of Murray-von Neumann equivalence of two projections in a von Neumann algebra? 

The basic intuition is that the equivalence of projections reflects an algebraic way of characterizing when two projections in $B(H)$ have the same rank. 
So one has an abstract notion of "rank", which works without the underlying substrate of the Hilbert space. Murray and von Neumann were very successful in using this idea to study II$_1$-factors. 
The idea is so powerful that it resurfaces in K-theory. 
