In how far are CCR and GCR C*-algebras interesting?

c-star-algebras operator-algebras

  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 Hausdorff. 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.

Related

prove $f(x)$ has at least $2n$ roots
Indefinite integral. Where is the mistake?
Practical applications of chaos theory in engineering or physics
Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$
Dual of a rational convex polyhedral cone
Relation between exterior (second) derivative $d^2=0$ and second derivative in multi-variable calculus.
What is the intersection of all Sylow $p$-subgroup's normalizer?
A question on vector space over an infinite field [duplicate]
Given $BA$, find $AB$.
Convert Semidefinite program forms
Learning to understand proofs faster?
Finite Element Method Weak Formulation