If A and B are C^\*-algebras that are algebraically isomorphic to each other, does 
this imply that they are \*-isomorphic to each other?



Are algebraically isomorphic $C^*$-algebras $*$-isomorphic?
Answering the question in the body of the original post, which seems to be more restricted than the implicit question in the title of the post....


The answer is YES. See



> 
> L. Terrell Gardner, *On isomorphisms of $C^\ast$-algebras.*
>  Amer. J. Math. **87** (1965) 384–396.
>  [MathReview](https://mathscinet.ams.org/mathscinet-getitem?mr=179637)
> 
> 
> 


Roughly speaking, the proof works by considering the ${\rm C}^\*$-algebras $A$ and $B$ as being represented on the GNS spaces $H\_A$ and $H\_B$ given by all pure states of $A$ and $B$ respectively, and then showing that an algebra isomorphism $A\to B$ can be extended to a spatial isomorphism ${\mathcal B}(H\_A)\to {\mathcal B}(H\_B)$.


Note that the original question was apparently raised by Sakai, a few years before Gardner's paper.


