Good reference for results about separable C* algebras I am studying the K-theory of C* algebras, and one result is of particular importance to me:

The $K_0$ group of separable C* algebras is countable

I found this answer that gives a good list of results about C* algebras and separability, does anyone knows a good reference for separable C* algebras? I would like to know:

*

*Necessary/ sufficient conditions for a c* algebra to be separable

*Properties of C* algebras

 A: There are very few instances where an object is called a C$^*$-algebra and it is not separable. With few exceptions, the only non-separable C$^*$-algebras that appear often in the literature are the infinite-dimensional von Neumann algebras; and these are usually treated as von Neumann algebras and not as C$^*$-algebras, for good reason: looking at these non-separable C$^*$-algebras as C$^*$-algebras is not very productive. As an example of this, $B(H)$ for infinite-dimensional separable $H$ is a non-separable, non-nuclear, non-amenable C$^*$-algebra; but as a von Neumann algebra it is separable, amenable, AFD, and as straightforward as a von Neumann algebra can be.
The zoo of separable C$^*$-algebras is so vast that it has been giving work to thousands of mathematicians for the last 80 years, and that is not about to change. The whole extremely successful Elliott Classification Program, that has been running for decades, is about classifying some separable C$^*$-algebras. Any C$^*$-algebra that is constructed out of some kind of generator set will be separable as long as the generator set is countable, which is (almost?) always the case with all C$^*$-algebras that are studied as such.
