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I want to have a better understanding of the Gelfand transform. This is a follow up regarding Julien's reply to the question on the spectrum of the sum of two commuting elements in a Banach algebra:

Assume that $A$ is commutative. Then by Gelfand, for every $x\in A$, we have $$ \sigma(x)=\{\phi(x)\;;\;\phi\in \hat{A}\} $$ where $\hat{A}$ denotes the set of characters (nonzero algebra homomorphisms from $A$ to $\mathbb{C}$).

It follows readily that for all $x,y\in A$: $$ \sigma(x+y)\subseteq\sigma(x)+\sigma(y)\qquad \mbox{and}\qquad\sigma(xy)\subseteq\sigma(x)\sigma(y). $$

My question is, why is clear from the above that we have

$$\sigma(x+y)\subseteq\sigma(x)+\sigma(y)\qquad \text{and} \qquad\sigma(xy)\subseteq\sigma(x)\sigma(y)?$$

I understand that $\phi$ are homomorphisms, but where do the spectrums take place in the definition and how did we get "$\subseteq$"?

Thank you.

Jessie
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Korn
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If $\lambda\in \sigma (x+y)$, then there is some $\varphi $ in $\hat A$, such that $\lambda =\varphi (x+y)$. Therefore $$ \lambda =\varphi (x+y) = \varphi (x)+\varphi (y)\in \sigma (x)+\sigma (y). $$ Similarly for $xy$.

Ruy
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