PlanetPhysics/Representations of Groupoids Induced by Measure

A groupoid representation induced by measure can be defined as measure induced [[../QuantumOperatorAlgebra4/|operators]] or as operators induced by a measure preserving map in the context of [[../QuantumOperatorAlgebra5/|Haar systems]] with measure associated with [[../LocallyCompactGroupoid/|locally compact groupoids]], ${𝐆}_{𝐥𝐜}$. Thus, let us consider a locally compact groupoid ${𝐆}_{𝐥𝐜}$ endowed with an associated Haar system $\nu =\{{\nu }^u,u\in U_{{𝐆}_{𝐥𝐜}}\}$, and $\mu$ a quasi-invariant measure on $U_{{𝐆}_{𝐥𝐜}}$. Moreover, let $(X_1,{𝔅}_1,{\mu }_1)$ and $(X_2,{𝔅}_2,{\mu }_2)$ be [[../LebesgueMeasure/|measure spaces]] and denote by $L^0(X_1)$ and $L^0(X_2)$ the corresponding spaces of [[../LebesgueMeasure/|measurable functions]] (with values in $ℂ$). Let us also recall that with a measure-preserving transformation $T:X_1⟶X_2$ one can define an operator induced by a measure preserving map , $U_T:L^0(X_2)⟶L^0(X_1)$ as follows.

\begin{displaymath} (U_T f)(x):=f(Tx)\,, \qquad\qquad f \in L^0(X_2),\; x \in X_1 \end{displaymath}

Next, let us define $\nu =\int {\nu }^{u}d\mu (u)$ and also define ${\nu }^{-1}$ as the mapping $x\mapsto x^{-1}$. With $f\in C_c({𝐆}_{𝐥𝐜})$, one can now define the measure induced [[../QuantumSpinNetworkFunctor2/|operator]] $In{d}^{‴}\mu (f)$ as an operator being defined on $L^2({\nu }^{-1})$ by the [[../Formula/|formula]]: $In{d}^{‴}\mu (f)\xi (x)=\int f(y)\xi (y^{-1}x)d{\nu }^{r(x)}(y)=f*\xi (x)$

Remark:

One can readily verify that :

$\Vert {\text{'}}^{″}In{d}^{‴}\mu (f)\Vert \le {\Vert f\Vert }_1,$

and also that $In{d}^{‴}\mu$ is a proper [[../CategoricalGroupRepresentation/|representation]] of $C_c({𝐆}_{𝐥𝐜})$, in the sense that the latter is usually defined for [[../QuantumOperatorAlgebra5/|groupoids]].