= Spectral idempotent from a separated spectrum
{title2=$e=\Theta_a(\mathbf1_X)$}
If a <spectrum of an element> splits into disjoint nonempty compact pieces $X,Y$, choose disjoint open neighborhoods and a <holomorphic function> equal to one on the first and zero on the second. The <holomorphic functional calculus> sends it to an <idempotent> with the corresponding zero/one values under every <algebra character>. It is neither zero nor identity: if one piece were killed, a reciprocal function on the other piece would make an originally spectral coordinate difference invertible.
Back to article page
