= Solution
The <Gelfand-Mazur theorem> says that a nonzero complex unital <Banach algebra> in which every nonzero element is invertible is the complex field itself: $a\mapsto\lambda$ where $a=\lambda1$. This is an isometric isomorphism under the usual normalization $\|1\|=1$; without that normalization its norm is simply $\|\lambda1\|=|\lambda|\|1\|$. We first prove the required spectral and resolvent facts rather than assume them.
For any element $a$ of a complex unital <Banach algebra> $A$, define its <spectrum of an element> by
$$
\sigma_A(a)=\{\lambda\in\mathbb C:\lambda1-a\text{ is not invertible}\}.
$$
If $\|b\|<1$, the partial sums of $\sum_{n\ge0}b^n$ converge in the complete algebra. Multiplying a partial sum by $1-b$ on either side gives $1-b^{N+1}\to1$, proving the <Neumann series> inverse. For $|\lambda|>\|a\|$ this gives
$$
R(\lambda,a)=(\lambda1-a)^{-1}=\sum_{n=0}^\infty\frac{a^n}{\lambda^{n+1}}.
$$
Thus the spectrum is bounded. Under $\|1\|=1$, $\|R(\lambda,a)\|\le(|\lambda|-\|a\|)^{-1}$; for an unnormalized identity the same estimate with numerator $\|1\|$ suffices. Also $\lambda R(\lambda,a)\to1$ as $|\lambda|\to\infty$.
If $\lambda_0$ is outside the spectrum and $R_0=R(\lambda_0,a)$, then for $|h|\|R_0\|<1$,
$$
R(\lambda_0+h,a)=\sum_{n=0}^\infty(-h)^nR_0^{n+1}.
$$
This follows by factoring $(\lambda_0+h)1-a=(\lambda_0 1-a)(1+hR_0)$ and applying the just-proved series. The series converges locally in norm, proving that the resolvent set is open, the spectrum is closed, and the <resolvent of an element> is holomorphic. Subtracting inverse equations also gives the <resolvent identity>
$$
R(\lambda,a)-R(\mu,a)=(\mu-\lambda)R(\lambda,a)R(\mu,a),
$$
consistent with the derivative $R'(\lambda,a)=-R(\lambda,a)^2$ from the local series.
The spectrum is nonempty. If it were empty, the resolvent would be an entire algebra-valued function tending to zero at infinity. Choose a <bounded linear functional> $\ell:A\to\mathbb C$ with $\ell(1)=1$, extending the functional on $\mathbb C1$ by the complex <Hahn-Banach theorem>. Then $\ell(R(\lambda,a))$ is an <entire function>, bounded outside a disk by the Neumann estimate and bounded inside by continuity and compactness. The <Liouville theorem> makes it constant, and its decay makes that constant zero. But $\lambda\ell(R(\lambda,a))\to\ell(1)=1$, a contradiction. We have proved that every spectrum is a nonempty compact subset of the disk $|\lambda|\le\|a\|$.
Now assume every nonzero element of $A$ is invertible. For any $a$, choose $\lambda\in\sigma_A(a)$. The noninvertible element $a-\lambda1$ must be zero. Thus every element is a scalar multiple of the identity, uniquely because $1\ne0$, and
$$
\boxed{A=\mathbb C1\cong\mathbb C.}
$$
No commutativity assumption was needed for this part; it is forced by the conclusion. Complex scalars are essential to the spectral argument.
For the <maximal ideal space> and its correspondence with <characters of an algebra>, now take $A$ to be a commutative complex unital <Banach algebra>. Every proper ideal lies in an algebraic <maximal ideal> by <Zorn's lemma>: a chain's union is an ideal not containing $1$. Every proper ideal also has proper closure. Otherwise it contains an element $m$ with $\|1-m\|<1$, which is invertible by the Neumann series, forcing the ideal to contain $1$. A maximal ideal $I$ is therefore closed, since its closure is a proper ideal containing it.
The quotient $A/I$, with its quotient norm, is a complex <Banach algebra>. Completeness follows from closedness of $I$, and submultiplicativity follows by taking infima over representatives. It is a division algebra: if $a\notin I$, maximality gives $I+Aa=A$, so $ba+i=1$ for some $b\in A$, $i\in I$. Hence $a+I$ is invertible. By the proved <Gelfand-Mazur theorem>, $A/I\cong\mathbb C$, with the identity mapping to $1$. Composing this isomorphism with the quotient map produces a nonzero multiplicative complex <linear functional> $\phi_I$ with kernel $I$.
Conversely let $\phi:A\to\mathbb C$ be a nonzero multiplicative complex linear functional, without assuming continuity. Nonzeroness and multiplicativity give $\phi(1)=1$, hence $\phi(\lambda1)=\lambda$ and surjectivity onto $\mathbb C$. Its kernel is a maximal ideal. If $a-\phi(a)1$ were invertible, applying $\phi$ to its product with an inverse would give $0=1$. Therefore $\phi(a)\in\sigma_A(a)$, so
$$
|\phi(a)|\le\|a\|.
$$
This proves <automatic continuity of characters>; with normalized identity their <operator norm> is one. Finally a kernel determines its character uniquely: for every $a$, its class in $A/I=\mathbb C1$ is a unique scalar multiple of the unit, and every character with kernel $I$ must take that scalar value. We have obtained the bijection
$$
\boxed{\{\text{maximal ideals of }A\}\longleftrightarrow\{\text{nonzero multiplicative complex linear functionals on }A\},\quad I\leftrightarrow\phi_I.}
$$
The commutativity in this correspondence cannot be dropped: $\operatorname{Mat}_2(\mathbb C)$ has the maximal two-sided ideal $0$, but has no scalar-valued character, since such a character would have zero kernel and give an impossible injective linear map into $\mathbb C$.
For use in the next question, the <spectrum equals character values in a commutative Banach algebra>. We have proved one inclusion. For the other, if $a-\lambda1$ is noninvertible, its generated ideal is proper, hence lies in a maximal ideal. The corresponding character takes $a$ to $\lambda$. Thus
$$
\sigma_A(a)=\{\phi(a):\phi\in\mathcal M_A\}.
$$
Give $\mathcal M_A$ the <Gelfand topology>, namely the <weak-star topology> of pointwise convergence on $A$ through these characters. It is Hausdorff and compact: the characters lie in the dual unit ball and form a weak-star closed subset, described by the closed equations $\phi(1)=1$ and $\phi(ab)=\phi(a)\phi(b)$. Apply the <Banach-Alaoglu theorem>. The <Gelfand transform> $a\mapsto\widehat a$, where $\widehat a(\phi)=\phi(a)$, is consequently a contractive unital algebra homomorphism into $C(\mathcal M_A)$; it separates distinct characters. This argument proves compactness and continuity, but does not assert injectivity or isometry for an arbitrary Banach algebra.
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