What is the right definition of "real von Neumann algebra"? Recall that a real C*-algebra is a Banach $\ast$-algebra $A$ over $\mathbb{R}$ which satisfies the standard C* identity and which also has the property that $1 + a^{\ast}a$ is invertible in the unitalization of $A$ for every $a$.  This is the "right" definition because the "real Gelfand-Naimark theorem" is true for such algebras: every real C*-algebra is isometrically $\ast$-isomorphic to a norm closed $\ast$-algebra of bounded operators on a real Hilbert space.
Now we turn to von Neumann algebras.  A von Neumann algebra is supposed to be a $\ast$-algebra of bounded operators on a (complex) Hilbert space which is closed in the weak topology, or equivalently the strong topology.  This can be abstracted to the intrinsic definition of a von Neumann algebra as a C* algebra which is the dual of some (complex) Banach space.  My question is: what is the intrinsic definition of a real von Neumann algebra which abstracts the notion of a $\ast$-algebra of bounded operators on a real Hilbert space which is closed in the weak topology or (equivalently?) the strong topology?
 A: Hi, this is intended as a comment on Jon's comment, but I still lack MO reputation to leave comments; sorry for that. I believe what is mentioned in Li's book is wrong; the right statement should be "a complex $C^\ast$-algebra is the complexification of a real one if and only if it has an involutory ${}^\ast$-antiautomorphism" (here an example by V. Jones of a von Neumann algebra antiautomorphic to itself but without involutory antiautomorphisms: http://www.mscand.dk/article.php?id=2523).  
Conversely, one can study real $C^\ast$-algebras in terms of their complexifications: e.g., say that $A$ is a real von Neumann algebra if $A\otimes_{\mathbb{R}}\mathbb{C}$ is von Neumann, and so on.
A: Real operator algebras have been studied by Bingren Li and others. Here's a paper of Li on the topic:
http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0936-8.pdf
I hope this helps!
A: Pedersen abstractly characterized von Neumann algebras as AW*-algebras with a separating family of completely additive states.  See here.
There's a notion of real AW*-algebra: see Berberian's text Baer $*$-rings, exercise 5.14.  Since an AW*-algebra is a C*-algebra in which the right annihilator of every set is generated by a single projection, whatever your definition of real C*-algebra is, this condition could easily be added to define a real AW*-algebra.
Thus, one could define a real von Neumann algebra to be a real AW*-algebra with a separating family of completely additive (real-valued?) states.
(Disclaimer: I'm no expert here, so take my comments with a grain of salt.)
A: A real von Neumann algebra (or real $W^*-$algebra) is a real *-algebra $\mathcal{R}$ of bounded linear operators on a complex Hilbert space containing the identity operator $\bf 1$, which is closed in the weak operator topology and satisfies the condition $\mathcal{R}\cap i\mathcal{R}=\{0\}$.
