Von Neumann algebras as reduced twisted groupoid $C^*$-algebras

Abstract

We characterize the von Neumann algebras that are isomorphic, as $C^*$-algebras, to reduced twisted $C^*$-algebras of locally compact Hausdorff groupoids equipped with continuous Haar systems of full support. They are exactly the subhomogeneous von Neumann algebras, equivalently finite products of matrix algebras over abelian von Neumann algebras. The groupoid can always be chosen compact, principal, and étale, with trivial twist and counting Haar system. We also prove that, for a groupoid in this class, the full or reduced twisted algebra is unital if and only if the groupoid is étale with compact unit space. This criterion reduces the classification for general Haar systems to the étale case. A further obstruction comes from controlled propagation, defined through faithful representations into the norm closure of uniformly sparse matrices. Every reduced twisted étale groupoid algebra and its Borel completion have controlled propagation, including for non-Hausdorff groupoids with locally compact Hausdorff unit space. For every infinite-dimensional Hilbert space $H$, every $*$-homomorphism from a nonzero quotient of $B(H)$ into an algebra with controlled propagation is zero. In particular, neither $B(H)$ nor the Calkin algebra embeds into any of these reduced or Borel groupoid algebras. We also prove that every von Neumann algebra with controlled propagation is finite. No separability or countability assumptions are required.