# Minimal projections vs maximal left ideals

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

First of all, let us note that the minimal projections in $zA^{**}$ are just minimal projections in $A^{**}$, because as you said $z$ is the supremum of minimal projections.
Let $L$ be a maximal closed left ideal in $A$. As you mentioned, the weak star closure of $L$ denoted by $L^{\circ\circ}$ is in the form of $A^{**}e$ for some projection in $A^{**}$ (indeed $e$ is just the weak star limit of the right approximate identity of $L$ and so plays the role of right unit for $L^{\circ\circ}$). You want to show that $1-e$ is a minimal projection. Assume that there is a non trivial projection $f\in A^{**}$ with $1-f\leq1-e$. Then $f-e$ is a projection with $L^{\circ\circ}(f-e)=0$. There is a net $\{a_i\}$ in $A$ converging to $f-e$ in the weak star topology ($A$ is embedded in $A^{**}$). One may check that $L$ doe son contain $\{a_i\}$ and the closed left ideal generated by $L$ and $\{a_i\}$ is still a proper in $A$. This contradictions shows that $1-f$ should be $0$ which implies that $1-e$ is minimal.