Let $A,B \subset B(H)$ be two concrete von Neumann algebra. Is $A\cap B$ a von Neumann algebra, too?
What about the intrinsic analogy of this question, as follows:
Let $C$ be a $C^*$ algebra and $A,B \subset C$ be two von Neumann algebras. Is their intersection, a von Neumann algebra, too?
Can one speak of a kind of minimal von Neumann algebra contained in a given $C^*$ algebra?
On the other extreme, can one think of a kind of maximal von Neumann algebra contained in a given $C^*$ algebra?
In particular what are two maximal von neumann algebras in $B(H)$ which are not isomorphic?