I'm working on a book on John von Neumann and need to gauge his importance in 21st century math. Any anecdotes or personal experience with his work and his legacy will be appreciated. Thanks.

Of all his work I am most familiar with von Neumann algebras, which play an important role in quantum mechanics, algebraic quantum field theory and noncommutative geometry. John von Neumann and Francis Murray were the first to develop these operator algebras in the 1930s and 1940s.

 

I don't know what his most important theorem is, but one that is quite remarkable is the von Neumann double commutant theorem. You can look up the exact result, but it is interesting because it relates purely topological aspects of certain operator algebras to purely algebraic properties. Have a look at nLab has to say.

I mentioned him in my reply on this post, I described how I viewed him.

 

http://www.scienceforums.net/topic/69319-entropy/

 

I think he was so far ahead in his thinking that a lot of his work is simply not understood or appreciated.

Von Neumann was one of the pioneers in the development of game theory.