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Homegroupoid representations induced by measure

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# groupoid representations induced by measure

# 1 Associated, Left or Right, Haar System

###### Definition 1.1.

A *groupoid representation induced by measure* can be defined as measure induced operator
or as an operator induced by a measure preserving map in the context of Haar systems with measure that are
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:

$(U_{T}f)(x):=f(Tx)\,,\qquad\qquad f\in L^{0}(X_{2}),\;x\in X_{1}$ |

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:

$\textbf{Ind}\mu(f)\xi(x)=\int f(y)\xi(y^{{-1}}x)d\nu^{{r(x)}}(y)=f*\xi(x)$ |

###### Remark 1.1.

One can readily verify that :

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

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.

## Mathematics Subject Classification

18D05*no label found*55N33

*no label found*55N20

*no label found*55P10

*no label found*55U40

*no label found*

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