A trace pairing and Elliott invariant for groupoid homology

Abstract

Abstract For an étale groupoid, we define a pairing between the Crainic–Moerdijk groupoid homology and the simplex of invariant Borel probability measures on the base space. The main novelty here is that the groupoid need not have totally disconnected base space, and thus the pairing can give more refined information than the measures of clopen subsets of the base space. Our principal motivation is upper C asterisk $C^*$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:mrow> </mml:math> -algebra theory. The Elliott invariant of a upper C asterisk $C^*$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:mrow> </mml:math> -algebra is defined in terms of upper K $K$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> -theory and traces; it is fundamental in the long-running programme to classify simple upper C asterisk $C^*$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:mrow> </mml:math> -algebras (satisfying additional necessary conditions). We use our pairing to define a groupoid Elliott invariant, and show that for many interesting groupoids it agrees with the upper C asterisk $C^*$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:mrow> </mml:math> -algebraic Elliott invariant of the groupoid upper C asterisk $C^*$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:mrow> </mml:math> -algebra: this includes irrational rotation algebras and the upper C asterisk $C^*$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>*</mml:mo> </mml:msup> </mml:mrow> </mml:math> -algebras arising from orbit breaking constructions studied by the first listed author, Putnam, and Strung. These results can be thought of as establishing a refinement of Matui’s HK conjecture for the relevant groupoids.