|B|^{-1} \sum_{x,y \in B} f(x)\overline{g(y)} = {\left\langle\Pi_{\mathcal{B}}f, g \right\rangle}.
\end{split}$$ This proves that $\Pi_\mathcal{B}$ is self-adjoint.

The first refinement property follows from the fact that $\Pi_{\mathcal{B}} f$ is $\mathcal{B}'$-measurable.

We utilise self-adjointness of $\Pi_{\mathcal{B}}$ and the first refinement property to conclude that $$\begin{split}
{\left\langle\Pi_{\mathcal{B}} f  , \Pi_{\mathcal{B}} f- \Pi_{\mathcal{B}'} f\right\rangle}& = {\left\langle\Pi_{\mathcal{B}} f  , \Pi_{\mathcal{B}} f-  f\right\rangle} = {\left\langlef  , \Pi_{\mathcal{B}} f-  \Pi_{\mathcal{B}} f\right\rangle} = 0.
\end{split}$$

Now we describe the particular type of factors that will be relevant to us.

\[local factor def\] A *simple real factor* of resolution $M$ is a factor of $[N]$ obtained by partitioning ${\mathbb{R}}$ into intervals all of length $M$.

A *simple congruence factor* of modulus $q$ is the factor of $[N]$ obtained by partitioning into congruence classes mod $q$.

We say that $\mathcal{B}$ is a *simple local factor* of resolution $M$ and modulus $q$ if it is the join of a simple real factor of resolution $M$ and a simple congruence factor of modulus $q$. Notice that $\mathcal{B}$ is a simple local factor if and only if it consists of the level sets of a local function (Definition \[local function def\]) of resolution $M$ and modulus $q$.

A *local factor* of dimension $d$, 