 C-star isomorphism

A C-star isomorphism is a bijective C-star homomorphism: a complex-linear multiplicative map preserving the involution. Between unital C-star algebras it automatically preserves the unit. It preserves the Banach algebra spectrum, since it preserves invertibility in both directions. Applying the C-star identity and the spectral norm formula to $a^{\ast }a$ gives $\Vert \Phi (a){\Vert }^2=\Vert \Phi (a^{\ast }a)\Vert =\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$, so it is automatically isometric.