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I have some basic familiarity with C*-algebras (from the mathematical physics side - Bratteli-Robinson), and while having a look at Arveson's "Invitation to C* algebras", I came across so-called CCR and GCR algebras (CCR unrelated to the mathematical physicists CCR-algebras). They seem to be somewhat known, I guess, however let me recall Arveson's definitions:

A CCR algebra is a C*-algebra $A$ such that, for every irreducible representation $\pi$ of $A$, $\pi(A)$ consists of compact operators.

Now, given a general C*-algebra $A$, for every irreducible representation $\pi$ (on some of $A$, we can consider the subset $\mathscr{C}_\pi:=\{x\in A| \pi(x)~\mathrm{compact~in~}\mathcal{B}(\mathcal{H})\}$, so basically the subset of all operators that are mapped to compact operators. The intersection of these sets for all irreducible representations is naturally a CCR-algebra called $\mathrm{CCR}(A)$. Now we define the GCR algebras:

A GCR algebra is a C*-algebra $A$ such that $\mathrm{CCR}(A/J)\neq 0$ for all ideals $J\neq A$.

Basically this means that the irreducible representations of C*-algebras contain all compact operators. In particular, finite dimensional C*-algebras are of this type, because all operators are compact.

Ok, for nonreflective Banach spaces, we should replace the "compact" with "completely continuous", which actually is the reason the CCR algebras are called like this, but I'm not really interested in this.

My questions are:

  1. Arveson claims that GCR algebras are in a sense the easy C*-algebras, because only there can we actually really study irreducible representations and write them down. Is there an intuitive reason, why this is the case? What do compact operators have to do with this?

  2. Also, is there a relation to the study of representations of von Neumann algebras? Here, we know that all irreducible representations are type I (if I'm not mistaken), but there are a lot of algebras with representations that are not type I and they are extremely important, e.g. in Quantum mechanics.

Or in short: What's the bigger picture behind these objects and are they still interesting objects to study?

Martin
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  • hi. I have a question about the definition of $CCR(A)$. Is $H$ a fixed Hilbert space for all irreducible represantions $\pi$, such that all $\pi(x)$ are in the same $B(H)$ for all $\pi$? Or depends $H$ on $\pi$ in the definition of CCR? I think $H$ is fixed, am I right? –  Jun 22 '16 at 09:18

1 Answers1

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  1. The fact that GCR C$^*$-algebras contain the compact operators implies that they have minimal projections; this makes them behave in a way that somehow resembles matrices. Also, their spectrum is $T_0$, and $T_1$ when they are CCR. These characteristcs made them easier to understand, and so in the 60s and 70s people studied them. So, it is not that much that they are interesting but rather they were the first ones that people could study.

  2. The intersection with von Neumann algebra theory is negligible. The double commutant of the image of a GCR through any representation is type I, and these have been completely understood since the 50s.

Finally, you'll be hard pressed to find a current paper in main C$^*$-theory that mentions them. They do appear though in related works.

Martin Argerami
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  • Can you give some intuition why the fact that they contain compact operators implies they have minimal projections? –  May 03 '20 at 20:25
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    The rank-one projections are minimal in $B(H)$, thus also minimal in any subalgebra. The rank-one projections are compact. – Martin Argerami May 03 '20 at 20:45
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    Very late for a party but I believe that the spectrum of GCR algebra need not to be Hausdorff (correct me if I'm wrong)-what can be said that it is always T_0 while for CCR is T_1. – truebaran Aug 23 '22 at 10:44
  • Indeed. Thanks for noticing! – Martin Argerami Aug 23 '22 at 11:29