Injectivity through a holomorphic functional calculus with nonvanishing derivative
= Injectivity through a holomorphic functional calculus with nonvanishing derivative
Let $x_1,x_2$ belong to a commutative unital <Banach algebra>, have the same <Gelfand transform>, and have common spectrum $K$. If $f$ is holomorphic near $K$ and $f'$ has no zero on $K$, then $f(x_1)=f(x_2)$ implies $x_1=x_2$. Apply the two-variable holomorphic functional calculus to the divided difference of $f$; its Gelfand transform never vanishes, so it is invertible.