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- W2285229441 abstract "For $pin (1,infty)$, we study representations of etale groupoids on $L^{p}$-spaces. Our main result is a generalization of Renault's disintegration theorem for representations of etale groupoids on Hilbert spaces. We establish a correspondence between $L^{p}$-representations of an etale groupoid $G$, contractive $L^{p}$-representations of $C_{c}(G)$, and tight regular $L^{p}$-representations of any countable inverse semigroup of open slices of $G$ that is a basis for the topology of $G$. We define analogs $F^{p}(G)$ and $F_{mathrm{red}}^{p}(G)$ of the full and reduced groupoid C*-algebras using representations on $L^{p}$-spaces. As a consequence of our main result, we deduce that every contractive representation of $F^{p}(G)$ or $F_{mathrm{red}}^{p}(G)$ is automatically completely contractive. Examples of our construction include the following natural families of Banach algebras: discrete group $L^{p}$-operator algebras, the analogs of Cuntz algebras on $L^{p}$-spaces, and the analogs of AF-algebras on $L^{p}$-spaces. Our results yield new information about these objects: their matricially normed structure is uniquely determined. More generally, groupoid $L^{p}$-operator algebras provide analogs of several families of classical C*-algebras, such as Cuntz-Krieger C*-algebras, tiling C*-algebras, and graph C*-algebras." @default.
- W2285229441 created "2016-06-24" @default.
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- W2285229441 date "2014-08-16" @default.
- W2285229441 modified "2023-09-23" @default.
- W2285229441 title "Representations of 'etale groupoids on $L^p$-spaces" @default.
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