# Sufficient conditions for a C* algebra to be separable

Do you know of any (necessary and) sufficient conditions for a C* algebra to be separable? Reference to bibliography is welcome.

I don't think there is a condition that is interesting. But it is easy to show that $$A$$ is separable if and only if it is generated by a finite or a countable set.
So as long as you have finite or countably many generators, your algebra will be separable. The vast majority of the commonly used C$$^*$$-algebras are like that. Exceptions would be the infinite-dimensional von Neumann algebras (and AW$$^*$$-algebras, but these are not as sexy).
The proof is extremely simple: if $$A$$ is separable, take $$X\subset A$$ countable and dense; then $$A=C^*(X)$$. Conversely, if $$X$$ is finite or countable and $$A=C^*(X)$$, then $$Y=\left\{\sum_{j=1}^m (a_j+ib_j)x_{1,j}\cdots x_{n,j}:\ n,m\in\mathbb N,\ a_j,b_j\in\mathbb Q,\ x_{k,j}\in X\cup X^*\right\}$$ is countable and dense.
• It's a two-liner, so I don't think there's a reference; I have included the argument. And yes, any type III von Neumann algebra is not separable as a C$^*$-algebra. They are usually separable as von Neumann algebras, though, when one uses the sot or wot topologies. – Martin Argerami Sep 25 '18 at 14:22