What is the right definition of "real von Neumann algebra"? - MathOverflow

What is the right definition of "real von Neumann algebra"?

2011-02-09T16:20:22Z

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?

Answer by Jon Bannon for What is the right definition of "real von Neumann algebra"?

2011-12-16T16:50:58Z

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!

Answer by Manny Reyes for What is the right definition of "real von Neumann algebra"?

2011-12-16T17:17:22Z

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.)

Answer by Mkouboi for What is the right definition of "real von Neumann algebra"?

2011-12-16T18:43:37Z

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.