I was reading a paper which mentioned without proof that every infinite-dimensional $C$* algebra has an infinite-dimensional commutative $C$* subalgebra.
Thinking about it for 10 minutes, I didn't see an immediately proof. It is sufficient to construct an element with infinite spectrum, but I don't see how to construct such an element.
Moreover, if one takes infinitely many noncommuting infinite and co-infinite projections on a Hilbert space, and takes the $C$* algebra generated by those projections, I don't see a reason why this contains an infinite-dimensional commutative $C$* subalgebra. (Clearly this is not a proven counterexample.)
Is there an easy proof of this fact?