Does every Hausdorff Compact space admits a (Radon) measure with full support?

Question

The title says it all. I know a bit of C*-algebra theory, so using Gelfand and Riesz theorems, this is equivalent to

Does every unital abelian C*-algebra admit a faithful state?

I believe this is true in case the C*-algebra is separable, which, if I remember correctly, is the case where the topological space (spectrum) is second countable: we can take a separating family $(\tau_n)$ of states on the C*-algebra and let $\tau=\sum_{n=1}^\infty 2^{-n}\tau_n$, which should do the job.

Any comment is welcome.

Answer 1

Try to find such a measure on the space $[0,\omega_1]$. The ordered set of ordinals up to the first uncountable ordinal $\omega_1$.