Vaibhav Kalvakota

Vaibhav Kalvakota

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Inception

2 minute read

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I am somewhat lame with writing things like this, but I think if I get right into it, it would save me from a lot of awkwardness that comes around with trying to explain why you are trying to explain things no one asked – but I bet people ask about von Neumann algebras (a really interesting thing), so I decided to try it this way.

Headings are cool – yes, I forgot to remove the header placeholder from the forked theme

I am currently working with a collaborator on a review on de Sitter things, like holography, quantum gravity, entanglement entropy and all that. One thing I realised, courtesy of my collaborator who was hell-bent on making use of vN algebras right from the first project we started working on, is that the entire backdrop of things like talking about Hilbert spaces, entanglement entropy and other things that are frequented by people working in, say AdS/CFT is built on this mathematical subject. For that matter, I was talking about the Ryu-Takayanagi formula without knowing why in the first place it makes sense to be able to define it in the first place; which is a bad thing to do, but in general this all seemed too abstract to properly get a hold on.

The notion of vN algebras in a naive perspective that caters solely to the childish need for a physical perspective in me is as follows: let $\mathcal{B}(\mathcal{H})$ be the space of all bounded operators in $\mathcal{H}$, and by $\mathcal{S}’$ we mean the commutant of $\mathcal{S}$. Then, a von Neumann algebra is a $*$-subalgebra $\mathcal{A}\subset \mathcal{B}(\mathcal{H})$ such that it satisfies $\mathcal{A}=\mathcal{A}’’$. Of course, I have not talked about weak operator topology, however it is absorbed in some sense, into the condition of the bicommutant. The intersection of $\mathcal{A}\cap \mathcal{A}’$ is also a von Neumann algebra, and the notion of factors is taken as usual. The classification of these factors into type I, II and III (we will not address the sub-decomposition as of yet into I${n}$, I${\infty }$ and so on, partly also because I am not sure how to put it into words at the moment), particularly the type III is fascinating, since this sums up the fact that a trace does not exist for certain von Neumann algebras. Adding more things in some time.

You can have many headings

Aren’t headings cool?

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