PlanetPhysics/Groupoid Representations Induced by Measure

A groupoid representation induced by measure can be defined as measure induced operators or as operators induced by a measure preserving map in the context of Haar systems with measure associated with locally compact groupoids, $\mathbf {G_{lc}}$. Thus, let us consider a locally compact groupoid $\mathbf {G_{lc}}$ endowed with an associated Haar system $\nu =\left\{\nu ^{u},u\in U_{\mathbf {G_{lc}} }\right\}$, and $\mu$ a quasi-invariant measure on $U_{\mathbf {G_{lc}} }$. Moreover, let $(X_{1},{\mathfrak {B}}_{1},\mu _{1})$ and $(X_{2},{\mathfrak {B}}_{2},\mu _{2})$ be measure spaces and denote by $L^{0}(X_{1})$ and $L^{0}(X_{2})$ the corresponding spaces of measurable functions (with values in $\mathbb {C}$). Let us also recall that with a measure-preserving transformation $T:X_{1}\longrightarrow X_{2}$ one can define an operator induced by a measure preserving map , $U_{T}:L^{0}(X_{2})\longrightarrow L^{0}(X_{1})$ 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 $f\in C_{c}(\mathbf {G_{lc}} )$, one can now define the measure induced operator $'''Ind'''\mu (f)$ as an operator being defined on $L^{2}(\nu ^{-1})$ by the formula: $'''Ind'''\mu (f)\xi (x)=\int f(y)\xi (y^{-1}x)d\nu ^{r(x)}(y)=f*\xi (x)$

Remark:

One can readily verify that :

$\left\|'''Ind'''\mu (f)\right\|\leq \left\|f\right\|_{1}$,

and also that $'''Ind'''\mu$ is a proper representation of $C_{c}(\mathbf {G_{lc}} )$, in the sense that the latter is usually defined for groupoids.