= Solution
Let $X$ be the <character space of an algebra>, consisting of nonzero multiplicative complex linear functionals on $A$. We construct the required <Gelfand transform> and prove its properties.
Characters exist. A maximal proper ideal exists by the axiom of choice and is closed: if it were dense it would contain an element within distance one of the identity, hence an invertible element, contradicting properness. The quotient is a complex unital Banach division algebra, and the <Gelfand-Mazur theorem> identifies it with $\mathbb C$, producing a character. The same argument applied to the proper ideal generated by $a-\lambda$ proves
$$
\sigma_A(a)=\{\chi(a):\chi\in X\}.
$$
Every character has norm one, since its value belongs to the spectrum and is bounded by $\|a\|$, while $\chi(1)=1$. The character equations are weak-star closed in the dual unit ball. Thus <Banach-Alaoglu theorem> makes $X$ compact Hausdorff; separability of $A$ makes it metrizable.
The algebra is inverse-closed in $B(H)$. If $a\in A$ is invertible there, $a^*a\geq cI$ for some $c>0$. For $M=\|a^*a\|$, the <Neumann series> for $(a^*a)^{-1}=M^{-1}[I-(I-a^*a/M)]^{-1}$ converges in $A$, since the positive contraction in brackets has norm at most $1-c/M<1$. Therefore $a^{-1}=(a^*a)^{-1}a^*\in A$. This proves <spectral permanence for C-star algebras> in the present concrete setting. In particular a self-adjoint element has real spectrum in $A$, so characters take real values on it. Decomposing $a$ into self-adjoint real and imaginary parts gives $\chi(a^*)=\overline{\chi(a)}$.
Define $\widehat a(\chi)=\chi(a)$. This is a unital star-homomorphism to $C(X)$. Every element of the abelian star-algebra is normal, and the <C-star identity> gives $\|a^{2^j}\|=\|a\|^{2^j}$. The <spectral radius formula> therefore gives $r_A(a)=\|a\|$. The displayed spectrum identity now yields $\|\widehat a\|_\infty=\|a\|$, proving isometry and injectivity. Its image is closed because $A$ is complete, contains the constants, is closed under conjugation, and separates distinct characters. The <Stone-Weierstrass theorem> makes that image dense in $C(X)$, hence equal to it. \b[$\boxed{A\cong C(X)}$ isometrically as a star-algebra.]
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