B(H) is not a twisted groupoid C*-algebra

Abstract

We show that $B(H)$ for an infinite dimensional Hilbert space $H$ cannot be realized as the reduced twisted $C^{\ast }$-algebra of any locally compact Hausdorff étale groupoid. The proof is based on the canonical conditional expectation $$C_r^{\ast }(G,\hat{I}\pounds )\to C_0(G^{(0)})$$ and a structural analysis of the resulting diagonal subalgebra inside $B(H)$. We show that this diagonal must be an atomic abelian von Neumann algebra, and then exclude both possibilities for its spectrum. If the unit space is finite, one obtains a tracial state on $C_r^{\ast }(G,\hat{I}\pounds )$, which is impossible for $B(H)$. If it is infinite, the groupoid structure forces a block-sparsity phenomenon for compactly supported sections, which is incompatible with $B(H)$. This provides the first examples of $C^{\ast }$-algebras that cannot be realized as reduced twisted étale groupoid $C^{\ast }$-algebras.

10 pages, v2: minor revision; added note in the introduction acknowledging independent work by David Gao; v3 is a revised version, to appear in Adv. Math