= Solution
A unital <C-star algebra> is a complex <Banach algebra> $A$ with a multiplicative identity $1_A$ and a conjugate-linear <involution> $a\mapsto a^*$ satisfying $(ab)^*=b^*a^*$ and the <C-star identity> $\|a^*a\|=\|a\|^2$. A <C-star isomorphism> $\Phi:A\to B$ is a bijective complex-linear map preserving products and the <involution>. It carries $1_A$ to $1_B$ and is automatically isometric.
For a commutative unital <C-star algebra>, let $\Omega$ be its <maximal ideal space>, equivalently the space of <Gelfand characters>, with the <weak-star topology>. A <Gelfand character> has $\chi(1)=1$ and $\|\chi\|=1$. The equations $\chi(ab)=\chi(a)\chi(b)$ and $\chi(1)=1$ define a closed in the <weak-star topology> subset of the dual unit ball. The <Banach-Alaoglu theorem> makes $\Omega$ a <compact Hausdorff space>. Maximal ideals give characters by the <Gelfand-Mazur theorem>, and the <Banach algebra spectrum> satisfies $\sigma(a)=\{\chi(a):\chi\in\Omega\}$.
The <Gelfand transform> is $\widehat a(\chi)=\chi(a)$. It preserves products, constants and the <involution>; the latter uses the fact that <characters of a C-star algebra respect the involution>. Every element of a commutative <C-star algebra> is normal, so the <C-star identity> and the <spectral radius formula> give
$$
\|\widehat a\|_\infty=\sup_{\chi\in\Omega}|\chi(a)|=r(a)=\|a\|.
$$
Thus its <image> is a closed self-adjoint subalgebra of $C(\Omega)$, contains the constants, and separates characters. The complex <Stone-Weierstrass theorem> makes that <image> dense and therefore all of $C(\Omega)$. This is the <Commutative Gelfand--Naimark theorem>.
For the <convergent sequence space> $c$, coordinatewise multiplication satisfies $\|\alpha\beta\|_\infty\leq\|\alpha\|_\infty\|\beta\|_\infty$, and coordinatewise complex conjugation is an <involution>. The constant sequence $1$ is the identity, and
$$
\|\alpha^*\alpha\|_\infty=\sup_n|\alpha_n|^2=\|\alpha\|_\infty^2.
$$
It is complete: a uniform limit of convergent sequences is convergent, so $c$ is a closed <linear subspace> of the <Banach space> $\ell_\infty$. Hence it is a commutative unital <C-star algebra>.
Take
$$
\boxed{\Omega=\mathbb N\cup\{\infty\},\quad\text{the one-point compactification of discrete }\mathbb N.}
$$
Every finite point is isolated, and the neighbourhoods of $\infty$ contain all but finitely many integers. This is a <compact Hausdorff space>: an open cover has a set containing $\infty$, leaving only finitely many points to cover. Define
$$
\Phi(\alpha)(n)=\alpha_n,\qquad\Phi(\alpha)(\infty)=\lim_{n\to\infty}\alpha_n.
$$
Continuity at $\infty$ is exactly convergence of the sequence. Thus $\Phi$ is onto $C(\Omega)$, and pointwise operations show it is a <C-star isomorphism> with $\|\Phi\alpha\|_\infty=\|\alpha\|_\infty$. Explicitly, these are all <Gelfand characters>: each $\chi(e_n)$ is $0$ or $1$ and at most one is $1$. If $\chi(e_n)=1$, multiplication by $e_n$ gives $\chi(\alpha)=\alpha_n$. If all vanish, <density of finitely supported sequences in l-p> in $c_0$ gives $\chi|_{c_0}=0$, so $\chi(\alpha)=\lim\alpha_n$.
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