arXiv subjects

Luiz Felipe Garcia

Publications and source records attributed to Luiz Felipe Garcia.

4 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

Operator algebras over the p-adic integers -- II

We continue the study of operator algebras over the $p$-adic integers, initiated in our previous work [1]. In this sequel, we develop further structural results and provide new families of examples. We introduce the notion of $p$-adic von Neumann algebras, and analyze those with trivial center, that we call ''factors''. In particular we show that ICC groups provide examples of factors. We then establish a characterization of $p$-simplicity for groupoid operator algebras, showing its relation to effectiveness and minimality. A central part of the paper is devoted to a $p$-adic analogue of the GNS construction, leading to a representation theorem for Banach $^*$-algebras over $\mathbb{Z}_p$. As applications, we exhibit large classes of $p$-adic operator algebras, including residually finite-rank algebras and affinoid algebras with the spectral norm. Finally, we investigate the $K$-theory of $p$-adic operator algebras, including the computation of homotopy analytic $K$-theory of continuous $\mathbb{Z}_p$-valued functions on a compact Hausdorff space and the analytic (non-homotopy invariant) $K$-theory of certain $p$-adically complete Banach algebras in terms of continuous $K$-theory. Together, these results extend the foundations of the emerging theory of $p$-adic operator algebras.

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arXiv2025

Operator algebras over the p-adic integers

We introduce $p$-adic operator algebras, which are nonarchimedean analogues of $C^*$-algebras. We demonstrate that various classical examples of operator algebras - such as group(oid) $C^*$-algebras - have nonarchimedean counterparts. The category of $p$-adic operator algebras exhibits similar properties to those of the category of real and complex $C^*$-algebras, featuring limits, colimits, tensor products, crossed products and an enveloping construction permitting us to construct $p$-adic operator algebras from involutive algebras over $\mathbb{Z}_p$. In several cases of interest, the enveloping algebra construction recovers the $p$-adic completion of the underlying $\mathbb{Z}_p$-algebra. We then discuss an analogue of topological $K$-theory for Banach $\mathbb{Z}_p$-algebras, and compute it in basic examples such as the $p$-adic Cuntz algebra and rotation algebras. Finally, for a large class of $p$-adic operator algebras, we show that our $K$-theory coincides with the reduction mod $p$ of Quillen's algebraic $K$-theory.

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