Let $X\subset\mathbb{R}^n$ be convex and open. We say that a distribution $T\in\mathcal{D}'(X\times X)$ is of positive type if $T(\bar{f}\otimes f)\geq 0$ for all $f\in\mathcal{D}(X)$. Now let $Y\subset X\times\mathbb{R}^n$ be given by $$ Y=\{(q,z)\in X\times\mathbb{R}^n:q\pm z\in X \} $$ and define a smooth map $\kappa: Y\to X\times X$ by $$ \kappa(q,z)=(q-z,q+z)\quad \forall (q,z)\in Y. $$ For this map the pullback $\kappa^* T\in\mathcal{D}'(Y)$ is well-defined. However, one notes that the set $Y$ is not a crossed product space (although it is a subset of $X\times \mathbb{R}^n)$.
My question is the following: In which sense can one say that the pullback $\kappa^*T$ of the distribution $T$ again is a distribution of positive type? An expression like $(\kappa^*T)(f\otimes g)$ does not make sense, since $Y$ is not a crossed product space. Rather, positivity of $T$ implies that $$ (\kappa^*T)((\bar{f}\otimes f)\circ\kappa^{-1})\geq 0\quad \forall f\in\mathcal{D}(X). $$ How is this related to the definition of positive definiteness of distributions stated above? Also, I'd like to know how it the above notion of positive definiteness is related to the definition of distributions of posiyive type in the context of the Bochner-Schwartz theorem (see e.g. Reed/Simon: Methods of modern mathematical physics II, Chapter IX.2), assuming that the distribution $T$ is not translation invariant, i.e $T(x,y)\neq T(y-x)$.