*-algebra, isomorphic to a C*-algebra

Question

A C-*-algebra A is *-isomorphic to a *-algebra B. Any examples when in this case B is not A C-*-algebra?

Answer 1

No, $B$ needs not be a $C^*$-algebra in general. Just take $A=B$ with a $C^*$ norm on one side and a non $C^*$-norm on the other side.

But note the following more interesting fact. If $A$ and $B$ are $*$-isomorphic $C^*$-algebras, then the $*$-homomorphism is automatically isometric.

This follows from the fact that $$ \|x\|^2=\|x^*x\|=\rho(x^*x) $$ where $\rho(x^*x)$ is the spectral radius of $x^*x$, which is algebraic and does not depend on the $C^*$-norm.

Answer 2

The question isn't well-defined. A precise question would be: If $A$ is a $C^*$-algebra, $B$ is a $*$-algebra and the underlying $*$-algebra of $A$ is isomorphic to $B$, does it follow that $B$ is the underlying $*$-algebra of a $C^*$-algebra? And is this $C^*$-algebra isomorphic to $A$?

The answer is yes. If $\phi : B \to A$ is a $*$-isomorphism, then $||-|| \circ \phi$ defines a norm on $B$, and it is easily checked that this defines a $C^*$-algebra structure on $B$. By construction $\phi$ is an isometric isomorphism of $C^*$-algebras.