Let $A$ be a factor von Neumann algebra. Then every element of $A$ can be writeen as a finite linear combination of projections in $A$. Is it right? I know a little about von Neumann algebra, thanks a lot to the one who tell me the answer or tell me where I can find the answer.
factor means $A\cap A'=\mathbb{C}I$. Thanks a lot!
On
The introduction to this article by Kaftal et al mentions several references that address this question.
In
Pearcy, C.; Topping, D., Sums of small numbers of idempotents, Mich. Math. J. 14, 453-465 (1967). ZBL0156.38102.
it is proved that such representation is possible in any properly infinite von Neumann algebra.