= Principal square root in a commutative Banach algebra
{title2=$y=\Theta_x(z\mapsto\exp(\tfrac12\operatorname{Log} z))$}
If the <spectrum of an element> $x$ lies in the open right half-plane, <holomorphic functional calculus> gives the displayed square root with spectrum in that half-plane. It is unique there. Two such roots $y,v$ have character values equal to the unique scalar square root with positive real part. Every <character of an algebra> is therefore nonzero on $y+v$, so the character description of the spectrum makes $y+v$ invertible. From $(y-v)(y+v)=0$ one obtains $y=v$. Equal character values alone do not prove equality in a possibly nonsemisimple algebra.
Back to article page
