Let $A$ be a unital abelian Banach algebra. Give me an example that two following inclusion relations is not true for all Banach algebras
$$\sigma(a+b) \subseteq \sigma(a)+\sigma(b) \forall a,b \in A$$
$$\sigma(ab) \subseteq \sigma(a)\sigma(b) \forall a,b \in A$$
Actually, these inclusions do always hold for elements $a$ and $b$ of a unital commutative Banach algebra. (If you take away commutativity, you can find counterexamples with 2-by-2 matrices.)
You can see this by applying the Gelfand transform $\Gamma:A\to C(X)$, with $X$ the maximal ideal space of $A$, because for all $a\in A$, $\sigma(a)=\Gamma(a)(X)$. The inclusions $(f+g)(X)\subseteq f(X)+g(X)$ and $(fg)(X)\subseteq f(X)g(X)$ are clear for arbitrary functions $f,g:X\to\mathbb C$.