Abstract
Given a second-countable, Hausdorff, étale, amenable groupoid \(\mathcal{G}\) with compact unit space, we show that an element a in \(C^{\star}(\mathcal{G})\) is invertible if and only if \(\lambda_{x}(a)\) is invertible for every x in the unit space of \(\mathcal{G}\),where \(\lambda_{x}\) refers to the regular representation of \(C^{\star}(\mathcal{G})\) on \(l_{2}(\mathcal{G}_{x})\). We also prove that, for every a in \(C^{\star}(\mathcal{G})\), there exists some \(x\;\in\;\mathcal{G}^{(0)}\) such that \(\parallel a \parallel\,=\,\parallel \lambda_{x}(a) \parallel\).
Mathematics Subject Classification (2010). Primary 22A22; Secondary 46L05, 46L55.
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Exel, R. (2014). Invertibility in Groupoid C*-algebras. In: Bastos, M., Lebre, A., Samko, S., Spitkovsky, I. (eds) Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, vol 242. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0816-3_9
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DOI: https://doi.org/10.1007/978-3-0348-0816-3_9
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