Question

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?

Answer 1

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.

Answer 2

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