Given a C*-algebra without unit.

Does there exist a nontrivial proper ideal
that does not lie in a maximally nontrivial proper ideal?

(For the unital case this follows easily by Zorn's lemma.)

Whatwould be an example of such?

  • $\begingroup$ Why doesn't Zorn's lemma apply here in the same way? $\endgroup$ – Omnomnomnom Jun 15 '16 at 17:47
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    $\begingroup$ @Omnomnomnom: Because a it may fail to be an inductive order. Crucial is the existence of a common element outside of every proper ideal, namely the identity. $\endgroup$ – C-Star-W-Star Jun 15 '16 at 17:50
  • $\begingroup$ Aha... interesting problem $\endgroup$ – Omnomnomnom Jun 15 '16 at 17:51
  • $\begingroup$ @Omnomnomnom: Thanks! :) I thought it would be nice to know what and how about it may go wrong in the non-unital case.. $\endgroup$ – C-Star-W-Star Jun 15 '16 at 18:03
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    $\begingroup$ See this answer: math.stackexchange.com/q/172215 $\endgroup$ – Prahlad Vaidyanathan Jun 22 '16 at 8:32

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