$\newcommand{\real}{\mathrm{real}}$I am having trouble with understanding the axiom (OS3) in [this book][1] by Glimm and Jaffe.
It defines
\begin{equation}
\mathcal{A} = \left \{ A(\phi) = \sum_{j = 1}^N c_j \exp \left( \phi(f_j) \right)  \; \Big \vert \; c_j \in \mathbb{C}, f_j \in \mathcal{D} \right \}
\end{equation}
where $\mathcal{D}$ is supposed to be the space of compactly supported, smooth functions on $\mathbb{R}^d$ and $\phi \in \mathcal{D}'(\mathbb{R}^d)$.
Now, this is notationally somewhat messed up (imho) but it is also stated earlier in the text, that $\mathcal{D}'(\mathbb{R}^d)$ should refer to *real* distributions hence, I assume that $\mathcal{D}$ should also refer to *real* functions - but I have not found this statement anywhere.

They then go on by defining the time-reflection operator $\theta : \mathcal{A} \to \mathcal{A}$ and consider a special supspace $\mathcal{A}_+ \subset \mathcal{A}$ and give the axiom:
\begin{equation}
0 \le \int_{\mathcal{D}'(\mathbb{R}^d)} \left( \theta A \right)^- A \mathrm{d} \mu
\end{equation}
where the superscript $-$ is supposed to mean complex conjugation.
This might sound okay but then they state that this is equivalent to the positive definiteness of matrices of the form
\begin{equation}
M_{i j} = \int_{\mathcal{D}'(\mathbb{R}^d)} \exp \left[ i \phi \left( f_i - \theta f_j \right) \right] \mathrm{d} \mu \left( \phi \right)
\end{equation}
with $f_j$ in the new space $\mathcal{D}_{\real}(\mathbb{R}^d)$.

I feel that it is completely unclear whether they are talking about $\mathcal{D}$ as a space of real or complex functions.
Furthermore, I cannot see the equivalence of the two statements unless $\mathcal{A}$ was defined with an additional $i$ in the exponent and with $\mathcal{D} \supset \mathcal{D}_{\real}(\mathbb{R}^d)$ as a space of *real* functions.

[nlab][2] also gives the almost identical definition as the book but without stating anything about real or complex spaces.


  [1]: https://link.springer.com/book/10.1007/978-1-4612-4728-9
  [2]: https://ncatlab.org/nlab/show/Osterwalder-Schrader+theorem