PlanetPhysics/Representations of Groupoids 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.