Pedro, could it be that in the Dirac construction, as it is written, the anticommutator $\left\{\psi(f),\psi^*(g)\right\}=\text{Re}\langle f , g \rangle \textbf{1}$? Instead, in order to get the inner product on th RHS of the anticommutator, may be one can take the conjugate (pre)Hilbert space $\overline{\mathfrak{k}}$ in the second slot, I mean $\mathfrak{h}=\mathfrak{k} \oplus \overline{\mathfrak{k}}$ ? Also this would make $f \mapsto \psi(f)$ anti-linear.