Involution-preserving ring isomorphisms in norm between unital $C^{\ast }$-algebras

Let $A$ and $B$ be nonzero unital $C^{\ast }$-algebras with units $1_A$ and $1_B$, respectively, and let $T\text{\hspace{0.17em}⁣}:A\to B$ be a bijection satisfying $$\mathrm{\parallel }T(a+b)\mathrm{\parallel }=\mathrm{\parallel }T(a)+T(b)\mathrm{\parallel },\mathrm{\parallel }T(ab)\mathrm{\parallel }=\mathrm{\parallel }T(a)T(b)\mathrm{\parallel },T(a^{\ast })=T(a{)}^{\ast }$$ for all $a,b\in A$. We prove that there exist a central symmetry $u$ in $B$ and a real $\ast$-isomorphism $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:A\to B$ such that $$T(a)=u\mathrm{\Phi }(a)(a\in A)\mathrm{.}$$ Moreover, $u$ and $\mathrm{\Phi }$ are uniquely determined by $T$. Conversely, if $u\in B$ is a central symmetry and $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:A\to B$ is a real $\ast$-isomorphism, then $T=u\mathrm{\Phi }$ is a bijection satisfying the three identities above. In particular, the two norm identities, together with involution preservation, force the normalized map $uT$ to preserve the full product, not merely the Jordan product.