# Can the condition that two von Neumann algebras are isomorphic to each other be characterized algebraically?

A *-algebra of operators on a given Hilbert space is a von Neumann algebra if it's equal to its double-commutant. That's a nice purely algebraic way of characterizing von Neumann algebras on a given Hilbert space.

Now suppose that $$M_1$$ and $$M_2$$ are two von Neumann algebras on the same separable Hilbert space $${\cal H}$$ over $$\mathbb{C}$$. Is there a purely algebraic way of expressing the condition that $$M_1$$ and $$M_2$$ are isomorphic to each other as von Neumann algebras, without explicitly referring to topology?

Here are examples of algebraic conditions that don't work (as far as I know), but they illustrate what I mean by "algebraic":

• Suppose that $$M_1=U^{-1}M_2 U$$ for some unitary operator $$U$$ on $${\cal H}$$. That's an algebraic condition, but it's not general enough, because two von Neumann algebras can be isomorphic to each other without being unitarily equivalent to each other.

• Suppose that the two von Neumann algebras $$M_1$$ and $$M_2$$ are isomorphic as *-algebras. That's an algebraic condition, but I doubt that it's specific enough, because I don't see any reason why *-isomorphism would imply isomorphism as von Neumann algebras.

Clarification: A comment pointed out that there are two notions of isomorphisms of von Neumann algebras: spatial and abstract. I am interested in abstract isomorphisms.

• There are two notions of isomorphisms of von Neumann algebras: spatial isomorphisms (your first bullet point) and abstract ones (your second bullet point). If neither of these is what you want, you should be say what it is instead. Jul 8, 2020 at 19:52
• In any case, $\ast$-isomorphisms between von Neumann algebras are necessarily isometric (preserve the norm) and normal (preserve the weak$^\ast$ topology). Maybe that was the piece you were missing when you discarded them in your second bullet point. Jul 8, 2020 at 19:53
• @MaoWao I edited the question to clarify that I'm interested in abstract isomorphism, and I think your second comment answers my question. I didn't realize that *-isomorphisms automatically preserved those other properties. I'll look into that and try to understand why it's true. Thank you! Jul 8, 2020 at 21:16

Based on hints in MaoWao's comment, I found https://math.vanderbilt.edu/peters10/teaching/spring2013, which includes the notes vonNeumannAlgebras.pdf. Corollary 1.3.2 in those notes confirms that a *-isomorphism between two von Neumann algebras is automatically isometric, and corollary 4.2.4 confirms that it automatically preserves the relevant topology. (See the top of page 32 for the definition of a key term used in the latter corollary.)