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

Alcides Buss, Luiz Felipe Garcia, Tomás Pacheco

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 étale groupoid. The proof is based on the canonical conditional expectation $$C_r^*(G,Σ)\to C_0(G^{(0)})$$ 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 $C_r^*(G,Σ)$, 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 étale groupoid $C^*$-algebras.

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

Abstract

We show that for an infinite dimensional Hilbert space cannot be realized as the reduced twisted -algebra of any locally compact Hausdorff étale groupoid. The proof is based on the canonical conditional expectation and a structural analysis of the resulting diagonal subalgebra inside . 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 , which is impossible for . If it is infinite, the groupoid structure forces a block-sparsity phenomenon for compactly supported sections, which is incompatible with . This provides the first examples of -algebras that cannot be realized as reduced twisted étale groupoid -algebras.
Paper Structure (12 sections, 15 theorems, 59 equations)

This paper contains 12 sections, 15 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Faithful expectations and von Neumann subalgebra
  3. Faithful expectations and atomicity
  4. Finite unit space and traces
  5. Excluding the infinite atomic case
  6. Atoms and corners
  7. Compact support implies block sparsity
  8. The sparse classes $S_k$
  9. A spreading operator
  10. Conclusion of the infinite case
  11. Proof of the main theorem
  12. Open questions

Key Result

Theorem 1.1

Let $H$ be an infinite-dimensional Hilbert space. Then there is no locally compact Hausdorff étale groupoid $G$ and no twist $\Sigma$ over $G$ such that

Theorems & Definitions (32)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 22 more