the spectral radius in a unital Banach algebra

Question

If x,y are commuting elements in a unital Banach algebra,how to prove that

$r(x+y)\leq r(x)+r(y)$and $r(xy)\leq r(x)r(y)$,where r is the spectral radius

Answer 1

Define $B$ to be the Banach subalgebra of $A$ generated by $\{1, x, y\}$. First you have to show that $r_A(a)=r_B(a)$ for all $a\in B$, where $r_A(a)$ denotes the spectral radius of $a$ in $A$ and similarly for $r_B(a)$.

Now you can use the Gelfand theory for the commutative Banach algebra $B$. In other words, use the fact that $r_B(a)=\|\hat{a}\|_{\sup}$, where $\hat{a}$ is the image of $a$ under the Gelfand transform; $a\mapsto \hat{a}$, $\hat{a}(\omega):=\omega(a)$ for all $\omega \in \Omega(B)$.