Inseparable $C^*$-algebras

Question

In many theories about $C^*$-algebras, the $C^*$-algebras are always assumed to be separable. I have a question: Why few people discuss the inseparble $C^*$-algebras? Are they more difficult to handle?

Answer 1

It really depends on what you are doing with your $C^*$-algebras. I wouldn't say that few people study nonseparable $C^*$-algebras, but a lot of modern areas of research (in particular, classification) are currently dealing with separable algebras, because, in short, they are easier to deal with (see my last paragraph). There isn't one all-encompassing reason why one would consider separable algebras over nonseparable algebras, but there are several reasons that come to mind. Here's a short list of possible reasons:

1. For a separable $C^*$-algebra, one can guarantee the existence of a sequential approximate unit (of course, if this is the only reason, you can just assume your algebra is $\sigma$-unital).
2. For separable $C^*$-algebras, there is always a faithful representation on a separable Hilbert space, and the space of bounded operators on separable Hilbert spaces is more well-behaved than the space of operators on nonseparable Hilbert spaces.
3. Any $C^*$-algebra is an inductive limit of its separable $C^*$-subalgebras. So if a given property passes to inductive limits, it suffices to check it on separable subalgebras.
4. Many naturally occurring classes of $C^*$-algebras are separable (examples in no particular order:$C(X)$ for $X$ compact and metrizable, group $C^*$-algebras of countable groups, AF-algebras, AF-embeddable algebras, AT-algebras, rotation algebras, Cuntz algebras, $\ldots$).

In regards to your last question, generally speaking, anytime someone adds an adjective to the class of objects they're studying, it's to make the problem simpler/ more accessible. You remove adjectives (or put "non" in front of them) to make the problem harder.