groupoid representations induced by measure

Definition 0.1   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.

$\displaystyle (U_T f)(x):=f(Tx)\,, \qquad\qquad f \in L^0(X_2),\; x \in X_1
$

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 $ \textbf{Ind}\mu (f) $ as an operator being defined on $ L^2(\nu^{-1})$ by the formula:

$\displaystyle \textbf{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 :

$\displaystyle \left\Vert \textbf{Ind}\mu(f) \right\Vert \leq \left\Vert f \right\Vert _1 $

,

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



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