Does every abelian C* algebra have a single self-adjoint generator?

Question

Does an abelian von Neumann algebra have this property? Is there some interesting class of C* algebras that does?

Answer 1

No for $C*$ algebras. If $A$ has a single generator then the maximal ideal space is a subset of $\Bbb C$. So $A=C(K)$ is a counterexample, if $K$ is a compact Hausdorff space not homeomorphic to a subset of $\Bbb C$.

Answer 2

Every abelian von Neumann algebra with separable predual is singly-generated: this is a result from von Neumann himself; there is a nice proof in Davidson's C$^*$-Algebras by Example.

For abelian C$^*$-algebras, as David mentioned in his answer, the singly-generated unital ones are precisely those $C(K)$ with $K\subset\mathbb C$ compact.