Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Consider (abstract) von Neumann algebras topologised by their weak*-topology arising from the unique predual.

In the theory of topological vector spaces, there is a natural notion of an inductive limit: recall that $E$ is an inductive limit of a directed system $(E_\alpha, \beta_\alpha)$ when the limit vector space $E$ is endowed with the final topology making all the embeddings $\hat{\beta_\alpha} \colon E_\alpha \to E$ continuous.

Is there an internal definition of an inductivie limit of von Neumann algebras? By internal I mean, a construction which does not appeal to the WOT-closure of the limit *-algebra represented on some Hilbert space. For example, can we perform the construction of the $2^\infty$-factor as an inductive limit in the language of topolgical vector spaces?

share|improve this question

2 Answers 2

Yes, see Proposition 5.5 and Proposition 5.7 in Andre Kornell's paper Quantum Collections.

share|improve this answer

It sounds like you want a representation independent construction, but did you know that every von Neumann algebra has an essentially unique canonical representation (the "standard form")? This is done in Takesaki vol II. I guess it's pretty straightforward to take the direct limit of a system of standard forms and I suspect that should do what you want.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.