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

TL;DR

The paper proves that the algebra of bounded operators on an infinite-dimensional Hilbert space cannot be represented as a reduced twisted groupoid C*-algebra, highlighting fundamental structural limitations.

Contribution

It establishes the first examples of C*-algebras that cannot be realized as reduced twisted étale groupoid C*-algebras, using structural analysis of diagonal subalgebras.

Findings

B(H) cannot be realized as a reduced twisted groupoid C*-algebra.

Diagonal subalgebras in B(H) must be atomic abelian von Neumann algebras.

Groupoid structure imposes incompatible properties on B(H).

Abstract

We show that $B(H)$ for an infinite dimensional Hilbert space $H$ cannot be realized as the reduced twisted $C^*$-algebra of any locally compact Hausdorff \'etale groupoid. The proof is based on the canonical conditional expectation $$C_r^*(G,\Sigma)\to C_0(G^{(0)})$$ 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 $C_r^*(G,\Sigma)$, 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 $C^*$-algebras that cannot be realized as reduced twisted \'etale groupoid $C^*$-algebras.