If A and B are C^*-algebras that are algebraically isomorphic to each other, does this imply that they are *-isomorphic to each other?
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4$\begingroup$ Why the vote to close? If I recall correctly this is a non-obvious result of Gardner (Gardener?) -- the point is that the given isomorphism between A and B is not assumed to be a star-HM $\endgroup$– Yemon ChoiCommented Mar 7, 2018 at 2:29
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2$\begingroup$ Note to the OP: the title of your question seems to be asking something more general than what you actually ask in your question... $\endgroup$– Yemon ChoiCommented Mar 7, 2018 at 2:30
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1$\begingroup$ I only clicked because I thought the first word was "homophobic" and now I feel dumb $\endgroup$– LoupaxCommented Mar 7, 2018 at 13:27
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$\begingroup$ Since there has been no response: I have edited the title of the question to match the actual question that was asked. If you object to this, please leave a comment to explain what you actually meant to ask $\endgroup$– Yemon ChoiCommented Mar 15, 2018 at 4:29
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1 Answer
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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
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.
answered Mar 7, 2018 at 2:42