# union of group von neumann

If we have an increasing chain of group von Neumann algebras such as $L(G_1)\subseteq L(G_2)\subseteq\ldots$ what can we say about the weak closure of their union? Is it a group von Neumann algebra? Thanks very much. Roya

• To talk about "closure" you need an environment. Where would the union live? This is key to answer the question. – Martin Argerami Jan 8 '16 at 12:54
• Since we work with a chain of group von Neumann algebras, I think that they are all contained in a $B(H)$ for some Hilbert space $H$ – roya Jan 10 '16 at 5:09
• Dear Martin, I'm so glad to see your comment. I will appreciate for any help. – roya Jan 10 '16 at 5:14

It might be usefull to known that $$L(\bigoplus_i G_i)=\overline{\bigotimes}_i L(G_i).$$

I would start by looking up the universal properties of the direct sum and tensor products in the case of tracial von Neumann algebras.

Edit: My first suggestion didn't make much sense, Martin Argerami made a very good point. You do know however that each group von Neumann algebra $L(G_i)$ is represented on some Hilbert space $\mathcal{H}_i$. Could you build somehow a larger Hilbert space $\mathcal{H}$ on which you can represent each $L(G_i)$ naturally? On that spac e you can talk about the closure of stuff and the question makes sense.

Closely related to these questions is the following paper: http://arxiv.org/pdf/1411.2799.pdf

The first senctence in remark 2.23 implies the statement I made above, but this paper is much more general. I can't find an easy reference of this property though.

• Dear mathematician42 thank you very much. I actually cant understand what so you mean by $L(\sum G_i)$. – roya Jan 8 '16 at 12:15
• Would you mind give me a reference that I can find the equality $L(\oplus_i G_i)=\bar{\otimes}_i L(G_i)$. Tank you. – roya Jan 8 '16 at 12:19
• Dear Mathematician thank you very much. – roya Jan 11 '16 at 20:10