arXiv subjects

Tomás Pacheco

Publications and source records attributed to Tomás Pacheco.

3 recordsLinked to original sources

arXiv2026

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

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.

math.OA↗

arXiv2026

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

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.

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arXiv2025

On weakly amenable groupoids

In this work, we study groupoids and their approximation properties, generalizing both the definitions and some known results for the group case. More precisely, we introduce weak amenability for groupoids using the definition of the Fourier algebra given by Renault. We prove that weakly amenable groupoids are inner exact. We also generalize its algebraic counterpart, the CBAP. To do this we introduce the notion of a quasi Cartan pair $(B,A)$ and see that $(C_r^*(G),C_0(G^0))$ can be viewed as such. We then define what it means for a pair $(B,A)$ to have the CBAP. We introduce the Cowling-Haagerup constants associated to these approximation properties and prove that $Λ_{\text{cb}}(C_r^*(G),C_0(G^0)) \leq Λ_{\text{cb}}(G)$. We then study some classes of groupoids where we could achieve equality, that is, $Λ_{\text{cb}}(G) = Λ_{\text{cb}}(C_r^*(G),C_0(G^0))$. They are discrete groupoids and groupoids arising from partial actions of a discrete group $Γ$ on a locally compact Hausdorff space $X$.

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