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Let ‎‎$‎A$ ‎be a‎ ‎unital ‎abelian‎ ‎Banach ‎algebra. ‎Give ‎me ‎an ‎example ‎that two ‎following ‎inclusion ‎relations ‎is ‎not ‎true ‎for ‎all ‎Banach ‎algebras‎

$$\sigma(a+b) ‎\subseteq ‎\sigma(a)+‎\sigma(b) ‎‎‎‎‎‎\forall a,b ‎\in A‎‎‎‎$$ ‎‎‎‎

$$\sigma(ab) ‎\subseteq ‎\sigma(a)‎\sigma(b) ‎‎‎‎‎‎\forall a,b ‎\in A‎‎‎‎$$

Ali Qurbani
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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 $f,g:X\to\mathbb C$.

Jonas Meyer
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  • The above answer is not true. see the $C^*$-algebras and operator theory by Gerard J. Murphy, Chapter 1 - Exercise 5 – Ali Qurbani Jan 20 '13 at 15:33
  • @AliQurbani: I do not know what that exercise says. Will you please share what that is, precisely? – Jonas Meyer Jan 20 '13 at 19:08
  • Exercise 5 of Murphy 's Book : Show that $\sigma(a+b) ‎\subseteq ‎\sigma(a)+‎\sigma(b)$ and $\sigma(ab) ‎\subseteq ‎\sigma(a)‎\sigma(b)$ ‎‎‎‎‎‎$\forall a,b ‎\in A‎‎‎‎$. Also show that this is not true for all Banach algebras – Ali Qurbani Jan 20 '13 at 19:55
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    @AliQurbani: Thanks. I think you misunderstand (and you've left out context for the first statement). The stated inclusions hold when the algebra is assumed to be commutative. The inclusions do not generally hold when the algebra is not commutative, and as mentioned in the second sentence of my answer, you can find examples in the algebra of 2-by-2 matrices. It should be straightforward, through guess and check if nothing else, to get two 2-by-2 matrices $a$ and $b$ such that the inclusions do not hold. You may also want to notice the link I posted below your question. – Jonas Meyer Jan 20 '13 at 21:08
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    @AliQurbani: But of course if you have trouble finding an example with 2-by-2 matrices or other means, you are welcome to ask follow-up questions. – Jonas Meyer Jan 20 '13 at 21:10