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?‎

From this answer we know that
$$
r(x)=\inf\{\Vert x^n\Vert^{1/n}:n\in\mathbb{N}\}
$$
So from this post we get the answer for a).
The functions $f_n(x)=\Vert x^n\Vert^{1/n}$ are continuous and a fortiori upper semi-continuous. Hence from this post we know that their infimum (which is actually $r(x)$) is upper semi-continuous too. This answers b).
