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Can we apply the Gelfand-Mazur theorem to the space $GL_n(\mathbb{C})\cup\{0\}$ with operator norm?

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  • $\begingroup$ Do you mean the theorem on $C^*$-algebras that are fields? $\endgroup$ – Olivier Bégassat Jan 3 '15 at 12:14
  • $\begingroup$ @OlivierBégassat Probably not, I mean the one that says: Division commutative complex Banach algebras are isometrically isomorphic to $\mathbb{C}$. $\endgroup$ – Bati Jan 3 '15 at 12:42
  • $\begingroup$ I think I've managed to verify all the assumptions (submultiplicativity comes from the properties of the op. norm, $||I||=1$, it is a vector space). Though, I couldn't find this simple result anywhere, so I'm asking here where is the catch. $\endgroup$ – Bati Jan 3 '15 at 12:49
  • $\begingroup$ Well, fields are commutative division algebras... In any case, $GL_n(\Bbb C)\cup\lbrace 0\rbrace$ isn't an algebra. $\endgroup$ – Olivier Bégassat Jan 3 '15 at 12:49
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    $\begingroup$ It's simply not a vector space. $\endgroup$ – Olivier Bégassat Jan 3 '15 at 12:58
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No, we can't ($GL_n(\mathbb{C})\cup\{0\}$ is not a vector space).

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