banach algebra, the spectral radius let A‎ ‎be a‎ ‎banach ‎algebra with identity and $a \in A$.
the spectral radius $a$: $ r(a)= sup \{ \lambda :‎ ‎‎\lambda ‎\in‎ ‎\sigma(a)‎\}$
a) if $A$ is  a‎ ‎‎abelian banach algebra, ‎$‎a,b ‎\in ‎A‎$‎, will below terms be correct? ‎
‎$‎r(ab)‎ ‎\leq ‎r(a) ‎r(b) ‎\\ ‎r(a+b)‎ ‎\leq ‎r(a) ‎+r(b)‎$
b) ‎say  ‎that ‎the ‎map ‎$ ‎r:A‎\longrightarrow‎\mathbb{R}‎\\a‎\mapsto ‎‎ ‎r(a)‎ $‎ ‎is ‎upper ‎semicountinuous?‎

$\newcommand{fs}[1]{\lVert #1\rVert}$
Without loss of generality, suppose $A$ is a unital abelian Banach algebra (that is $\lVert 1\rVert=1$). Theorem of Gelfand Representation states that $r(a)=\lVert \hat{a}\rVert_{\infty}$(see Theorem 1.3.6 in Murphy's book  $C^*$-algebra and Operator Theory), Hence, \begin{align*}
&r(a+b)=\lVert\widehat{a+b}\rVert_\infty=\fs{\hat{a}+\hat{b}}_\infty\leq\fs{\hat{a}}_\infty+\fs{\hat{b}}_\infty=r(a)+r(b),\\
&r(ab)=\lVert\widehat{ab}\rVert_\infty=\fs{\hat{a}\hat{b}}_\infty\leq\fs{\hat{a}}_\infty\fs{\hat{b}}_\infty=r(a)r(b)
\end{align*}
My answer to (b) is as same as the other.
