= Principal cube root of a Banach-algebra element
{title2=$x^{1/3}=\exp(\tfrac13\operatorname{Log}x)$}
If the <spectrum of an element> $x$ avoids $(-\infty,0]$, its <holomorphic functional calculus> value for the <principal cube root> is the unique $y$ with $y^3=x$ and spectrum in $\{z\ne0:|\arg z|<\pi/3\}$. For uniqueness, any other such $v$ defines a <continuous> unital <algebra homomorphism> $f\mapsto\Theta_v(f\circ(z\mapsto z^3))$ from the slit-plane <holomorphic functions> to the algebra. It sends the coordinate to $x$, so <continuity and uniqueness of holomorphic functional calculus> identify it with $\Theta_x$. Applying it to the principal cube root gives $v=y$. This argument does not assume commutativity of the ambient algebra or separation by <algebra characters>.
Back to article page
