As we know, a minimal projection must be Abelian, An Abelian projection must be finite. A minimal projection correspond to a rank one operator, a finite projection correspond to a finite rank operator, what kind of operator be corresponded with the Abelian projection?

I find some book use Abelian projections to define the type I von-Neumann algebras, but others use minimal projections instead. In general, how close are minimal projections to Abeilian projections?


1 Answer 1


You cannot define type I von Neumann algebras in terms of minimal projections, since many don't have them; $L^\infty[0,1]$, for example. What you probably saw is that some texts only care about factors, and in a factor the abelian projections are precisely the minimal ones.

Rank one projections (not "operators") are an example of a minimal projection. But minimality is a condition inside the algebra. If you consider an infinite projection $p$ and let $M=\{\alpha p+\beta(1-p):\ \alpha,\beta\in\mathbb C\}$, then $M$ is a von Neumann algebra with $p$ a minimal projection; I wouldn't call it a "rank one operator".

The same with finite projections. Finite rank projections are the finite projections of the von Neumann algebra $B(H)$. But all projections in a II$_1$-factor are finite, and no one would call those "finite-rank".

As $B(H)$ is a factor, its only abelian projections are the minimal ones. So there is no natural operators that you would call "abelian projections".

  • $\begingroup$ Oh, yes, it is factor! In general, does any abelian projection can be decomposed into a minimal projection and a projection in an Abelian von Neumann algebra(each part maybe zero)? $\endgroup$
    – Strongart
    Aug 19, 2014 at 5:32
  • $\begingroup$ Sorry but I cannot really make sense of how you want to write your projection. What do you mean by "a projection in an Abelian von Neumann algebra"? $\endgroup$ Aug 19, 2014 at 5:42
  • $\begingroup$ Sorry, I do not get the standard word to say this ideal, roughly speaking, you give an example L∞[0,1] which has no minimal projections, the projection in it is an example, maybe we can call it "Abelian projection containing no minimal projection". $\endgroup$
    – Strongart
    Aug 19, 2014 at 14:03
  • $\begingroup$ I think I understand what you are saying. Suppose your vN algebra is $\mathbb C^2\oplus M_2(\mathbb C)$. Then $(1,1)\oplus 0$ is an Abelian projection, and $(1,0)\oplus0$ and $(0,1)\oplus 0$ are proper minimal projections below your Abelian projection. $\endgroup$ Aug 19, 2014 at 15:42
  • 1
    $\begingroup$ Probably because they have been called "factors" for the last 80 years. $\endgroup$ Aug 22, 2014 at 12:37

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