In precisely what sense is the universal representation of a C*-algebra “universal”

c-star-algebrasfunctional-analysisoperator-algebrasrepresentation-theory

Recently I started learning about the universal representation of a C*-algebra $A$, which is the direct sum the GNS representations corresponding to each state $\rho\in S(A)$. Let me denote it $\pi_u : A\to\mathcal{B}(H_u)$. Certainly it's remarkably useful to know that every C*-algebra is *-isomorphic to a norm-closed subalgebra of some $\mathcal{B}(H)$. However, I have yet to really see in what sense this representation is "universal". Why attach that qualifier?

The tag "universal" suggests this representation has some relationship to all other (faithful?) representations $\pi : A\to\mathcal{B}(H)$. The wikipedia entry on the universal representation says that if $\pi$ is any other representation, then there exists a "projection $P$ in the center of $\overline{\pi_u(A)}$ (all closures will be taken wrt the weak operator topology), and a *-isomorphism $\alpha : \overline{\pi_u(A)}P\to\overline{\pi(A)}$ such that
$$
\pi(a) = \alpha(\pi_u(a)P)\tag{*}
$$

Wikipedia provides no reference for this result, which annoys me as I'd like to try to understand a few things:

  • Where does this projection $P$ come from? Why is it necessary that it be in the center of $\overline{\pi_u(A)}$?
  • Is (*) the best we can do? The projection $P$ seems crucial, but it also muddles what would've otherwise been a simpler "universality" result, akin to the universality of, say, the Stone-Cech compactification.
  • In particular, I was trying to prove that if $\pi_u$ admits a weak expectation (a unital, completely positive contraction $\varphi : \mathcal B(H)\to\overline{\pi_u(A)}$ such that $\varphi(\pi_u(a)) = \pi_u(a)$ for all $a$), then $\pi$ also admits a weak expectation. The proof would've been straightforward, if not for the projection $P$, and I can't see how to "get around it". (I know now that you can prove this using Arveson's extension theorem, but $P$ still irks me).
  • Is there a sense in which the universal representation is functorial?

Can anyone help me understand what I'm missing?

Best Answer

The representation $\pi_u$ is universal in the sense that the von Neumann algebra $\overline{\pi_u(A)}$ is universal. This means that if $\pi$ is another representation, there exists a sigma-wot continuous $*$-epimorphism $\rho:\overline{\pi_u(A)}\to\overline{\pi(A)}$. In other words, $\overline{\pi(A)}$ is a quotient of $\overline{\pi_u(A)}$. With less symbols, any von Neumann algebra generated by $A$ is a quotient of $\overline{\pi_u(A)}$.

As for your bullet points:

  • The kernel of a $wot$ continuous $*$-homomorphism is a $wot$ closed ideal. These are given by compressions by central projections. That is, $\ker\rho=\overline{\pi_u(A)}\,P$ for a certain central projection $P$. You make it sound as if being central is a bad thing. The more properties you have for an object, the more determined it is. Knowing that $P$ is central lets you do manipulations that you wouldn't be able to do otherwise.

  • "Best we can do?" You seem to see a problem with this construction, when there isn't one. What would be "better"? Given $A$, there exists a universal von Neumann algebra such that the von Neumann algebra generated by any representation of $A$ is a quotient of it. That sounds very "universal" to me.

  • Trying to avoid Arveson's Extension Theorem here sounds like trying to avoid Hahn-Banach in the typical functional analysis' proofs (they are the same theorem, after all). The weak expectation for $\pi(A)$ will not have the same domain as $\varphi$ (as $\pi(A)$ lives in a different $B(H_\pi)$), so the natural argument is to work on the "common ground" and extend by Arveson's Extension Theorem. I'm not sure I understand what you mean when you say that the proof would have been straightforward "without $P$".

  • For the universal representation to be functorial you would need it to map between two categories. Not sure what the categories would be here (disclaimer: my category theory is weaker than weak).

For areference, look at Theorem III.2.4 in Takesaki I. Or Theorem 10.1.12 in Kadison-Ringrose II.

Edit: I forgot to mention that the canonical notation for $\overline{\pi_u(A)}$ is $\pi_u(A)''$. Or even $A''$ for some authors.

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