countable unitary element in a separable C*-algebra

Question

How do I show that the set of unitary equivalence classes of projections is countable in a unital separable $C^*$-algebra?

So I tried to show that the set of unitary elements in $C^*$-algebra is countable, but it was not successful.

Thanks.

Answer 1

First, a comment: every unital C$^*$-algebra has uncountably many unitaries (simply because $\mathbb C $ does).

For your problem, if the projections $p $ and $q $ are not unitarily equivalent, then $\|p-q\|\geq1$; see this. So, uncountably many pairwise non-unitarily equivalent elements would all be at distance $1$ from each other, contradicting separability.