# Infinite dimensional C*-algebra contains infinite dimensional commutitive subalgebra

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?

Now, apply the spectral theorem to such an element $x$. Using continuous functional calculus, you will get that $C^*(x)=C_0(\sigma(x))$, which is infinite-dimensional as $\sigma(x)$ is an infinite compact set.