Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.01802

The Baum-Connes conjecture: a concise course

Alain Valette
View original source ↗Read source document ↗Export metadata ↓

Abstract

These are the slightly edited notes from a 10-hour course I taught at Institut Henri Poincaré in January-February 2025 in the framework of the thematic semester "Representation theory and noncommutative geometry".

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alain Valette. 2026-10-01. The Baum-Connes conjecture: a concise course. https://arxiv.org/abs/2610.01802

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Hao-Ng isomorphism theorem for reduced crossed products

We prove the Hao-Ng isomorphism for reduced crossed products by locally compact Hausdorff groups. More precisely, for a non-degenerate $\mathrm{C}^*$-correspondence $X$ and a generalized gauge action $G \curvearrowright X$ by a locally compact Hausdorff group $G$, we prove the commutation ${\mathcal{O}}_{X\rtimes_rG}\cong {\mathcal{O}}_X\rtimes_rG$ of the reduced crossed product with the Cuntz-Pimsner C*-algebra construction. Our proof shows how such commutations with reduced crossed products can follow from more general principles in non-self-adjoint crossed product theory.

math.OA↗
arXiv2026

Operator Norm Bounds for Multi-leg Matrix Tensors and Applications to Random Matrix Theory

We study the extremal values of multi-leg traces of matrix tensors under operator norm constraints. A graphical representation gives upper and lower bounds expressed as powers of the common tensor-factor dimension. The lower bounds are attained by matrices that permute tensor factors and are exact within this family. We prove the bounds first for two tensor factors and then for an arbitrary number. Leaving some indices uncontracted produces matrices for which we obtain both moment and operator norm bounds; a single choice of coefficient matrices attains the lower bounds for every positive integer moment. We also obtain exact scalar and operator norm maxima in special cases with sufficiently many tensor factors sharing the same cyclic contraction. As an application, the operator norm bounds yield a uniform comparison between Ginibre products and their free circular counterparts with growing matrix coefficients.

math.OA↗
arXiv2026

On ultraproduct approximations and property (T) factors

We introduce a framework allowing for key aspects of deformation/rigidity theory to be used in the study of continuous model theory of II$_1$ factors. Using this framework, we solve several well-known open problems in the area. For example, we show that the group von Neumann algebras $L(\operatorname{SL}_3(\mathbb Z))$ and $L \mathbb F_2$ are not elementarily equivalent, and we show that the group von Neumann algebra $L\mathbb F_2$ is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II$_1$ factor. Building on previous work of Boutonnet, Chifan and Ioana, we also provide a continuum of pairwise non-elementarily equivalent full factors, which can be taken to be group von Neumann algebras.

math.OA↗