Groupoid algebras as Cuntz-Pimsner algebras

Abstract

<p>We show that if <i>G</i> is a second countable locally compact Hausdorff étale groupoid carrying a suitable cocycle <i>c</i>:<i>G</i> → ℤ, then the reduced <i>C</i>*-algebra of <i>G</i> can be realised naturally as the Cuntz-Pimsner algebra of a correspondence over the reduced <i>C</i>*-algebra of the kernel <i>G</i>₀ of <i>c</i>. If the full and reduced <i>C</i>*-algebras of <i>G</i>₀ coincide, we deduce that the full and reduced <i>C</i>*-algebras of <i>G</i> coincide. We obtain a six-term exact sequence describing the <i>K</i>-theory of <i>C</i>*_r (<i>G</i>) in terms of that of <i>C</i>*_r (<i>G</i>₀).</p>