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I've seen in some papers a statement (which is referred to a very old book of Dixmier in French which I have no access to / can't read anyway) saying that maximal left ideals of a (unital) C*-algebra $A$ are in one-to-one correspondence with minimal projections in the algebra $zA^{**}$, where $z$ is the central projection being the supremum over all minimal projections in $A^{**}$.

Could one please sketch how does this correspondence work?

EDIT: One way: take a maximal left ideal $L$. Then its bipolar $L^{\circ\circ}$ is a $\sigma$-closed maximal left ideal of $A^{**}$. Consequently, there is a minimal projection $e$ in $A^{**}$ such that $L^{\circ\circ}=A^{**}(1-e)$.

The other way round: Take a minimal projection $e$ in $zA^{**}$ and consider the left ideal $A^{**}(1-e)$. It seems that $A\cap A^{**}(1-e)$ is a maximal left ideal of $A$.

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For what it's worth: the books on operator algebras by Dixmier were all translated into English, as far as I know. – t.b. Aug 11 '12 at 20:05
@BrunoM. May be you should write this edit as answer to your question? – Norbert Aug 12 '12 at 19:16
@Norbert; this is not a full answer, since I have no idea how to associate a maximal left ideal of $A$ with a minimal projection in $zA^{**}$. – Tomek K. Aug 12 '12 at 19:31

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