Explicit example of Gel'fand transform I would like to determine explicitly the Gel'fand transform for the (commutative unital) Banach algebra of $2 \times 2$ matrices of the form $$\pmatrix{ a && b \\\ 0 && a}, \quad a,b \in \mathbb C.$$ Such matrices can evidently be decomposed into the sum of a multiple of the identity and a nihilpotent matrix, so it is clear that the Gel'fand transform is not isometric, since at least all matrices with $a=0$ and $b$ arbitrary are into the kernel (indeed, these ones have zero spectral radius). I would appreciate a detailed solution, for the sake of having an explict example. It would be of interest also if you can provide some references with detailed examples (say, in the spirit of Kadison-Ringerose volumes III and IV or even papers).

You first want to determine the characters. Since every element is of the form $aI+bE$, with $E$ nilpotent, it is easy to check that the only character is the map $\varphi:aI+bE\longmapsto a$. Indeed, write $\gamma=\varphi(E)$. Then $$ \varphi(aI+bE)=a+b\gamma. $$ For this to be multiplicative, we need \begin{align} \gamma^2&=\varphi\left(E\right)^2=\varphi (E^2)=\varphi (0)=0. \end{align} This forces $\gamma=0$, so $\varphi(aI+bE)=a$.

Now the Gelfand transform is the map $\Gamma:A\to C(\\{\phi\\})=\mathbb C$, with $$ \Gamma(aI+bE)(\varphi)=\varphi(aI+bE)=a. $$