An object $X$ of a given category is called projective if for each morphism $f : X \rightarrow Z$, and each epimorphism $ g : Y \twoheadrightarrow Z$, there is a morphism $h : X \rightarrow Y$ such that $f=gh$.

An ordered vector space $X$ is said to have the Hahn-Banach extension property if for each real vector space $Y$, each subspace $Z$ of $Y$, each sublinear operator $V : Y \rightarrow X$, and each linear operator $T : Z \rightarrow X$ satisfying $T \leq V_{|Z}$ pointwise, there is a linear operator $\hat T : Y \rightarrow X$ with $\hat T_{|Z}=T$ and $\hat T \leq V$.

A theorem of Gleason asserts that a compact (Hausdorff) space $K$ is projective in the category $\mathbf{CHaus}$ of compact Hausdorff spaces iff $K$ is extremally disconnected. By results of Goodner and Nachbin, the ordered vector space $C(K)$ has the Hahn-Banach extension property iff $K$ is extremally disconnected. It is thus known that: $K$ is projective in $\mathbf{CHaus}$ iff $C(K)$ has the Hahn-Banach extension property $(\star)$.

But : the Hahn-Banach extension property is a kind of projectivity with reversed arrows (reflecting the contravariance of the functor which sends maps $f : X \rightarrow Y$ to $u \in C(Y) \mapsto u \circ f \in C(X)$ ), so that it is reasonable to expect a "direct proof" of $\star$. It is relatively easy to show the direction "$\rightarrow$" in the equivalence (see below), and I would be very happy if the other direction also had a "direct proof" ... Thanks in advance.

Proof of the direction "$\rightarrow$" in $\star$ : Let $K$ be a projective object of $\mathbf{CHaus}$, and $(Y, Z, V, T)$ as in the definition of the Hahn-Banach property. Following Rainwater, let $D$ be the set $K$ endowed with the discrete topology and $r : \beta D \twoheadrightarrow K$ be the (unique) continuous extension of the canonical map $D \rightarrow K$ to the Stone-Cech compactification $\beta D$ of $D$. By projectivity, there is some $i : K \rightarrow \beta D$ with $id_{K} = r \circ i$. Without loss of generality $i$ is an inclusion map. Then $r$ is a retraction of $\beta D$ onto $K$. The maps $r$ and $i$ induce (by the functor described above) maps $\tilde r : C(K) \rightarrow C(\beta D)$ and $\tilde i : C(\beta D) \rightarrow C(K)$. As a consequence of the usual Hahn-Banach theorem, the space $C(\beta D) \simeq \ell^\infty(D)$ has the Hahn-Banach extension property ; applying this to $(Y,Z,\tilde r \circ V, \tilde r \circ T)$ yields an operator $\hat T'$ with $\hat T'_{|Z}=\tilde r \circ T$ and $\hat T'\leq \tilde r \circ V$. The desired operator is $\hat T = \tilde i \circ \hat T'$.