Example of norm separable c-star algebras

Question

I want to know enough examples of norm separable $C^{*}$-algebras which are neither finite dimensional nor commutative.

Answer 1

Your desire is somewhat vague since you do not write down your motivation.

Did you already browse through K. R. Davidson's book $\,C^*$-Algebras by Example ? (*)
It's worthwhile!

At least kinda start of a list, with $\mathsf H$ being an infinite-dimensional separable Hilbert space:

• Compact operators $\:\mathcal K(\mathsf H)$

• Cuntz algebras $\:\mathcal O_n$ with $\,n\in\{2,3,\dots,42,\dots ,\infty\}$

• $\dots$

_{* Fields Institute Monograph Volume 6, American Mathematical Society, 1996}

Answer 2

Any C$^*$-algebra which is generated by a finite or a countable set will be separable. Including the examples by Hanno, here is a very incomplete list:

• $K(H)$

• Cuntz and Cuntz-Krieger algebras

• For any countable non-abelian group $\Gamma$, the reduced C$^*$-algebra of the group, $C_r^*(G)$ (namely the norm closure of the span of $\lambda(G)\subset B(\ell^2(G))$, where $\lambda$ is the left regular representation). This includes for instance $C_r^*(\mathbb F_n)$, where $\mathbb F_n$ are the free groups.

• For any countable non-abelian group $\Gamma$, the universal C$^*$-algebra of the group, $C^*(G)$. This differs from the above whenever $G$ is non-amenable.

• Irrational rotation algebras ($A_\theta$ is the universal C$^*$-algebra generated by unitaries $u,v$ such that $uv=e^{2\pi i\theta}vu$)

• AF C$^*$-algebras. That is, direct limits of finite-dimensional C$^*$-algebras. This class is huge, as can be seen by considering their Bratelli diagrams

• Tensor products of the above

• Reduced free products of the above

• $c_0$-sums of the above

• C$^*$-subalgebras of the above