C-star isomorphism
= C-star isomorphism
{c}
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 $a^*a$ gives $\|\Phi(a)\|^2=\|\Phi(a^*a)\|=\|a^*a\|=\|a\|^2$, so it is automatically isometric.