Eva-Maria Hekkelman: DOIs & MOIs: pourquoi? Double Operator Integrals and Multiple Operator Integrals have a reputation of being quite technical objects of study in Functional Analysis, but in fact they appear in many places in noncommutative geometry. In this first part of the mini-course on MOIs I'll show the natural ways they arise, and explain two ways of constructing them. We will see that this gives us a handy tool for toying around with functional calculus and perturbations.

Pierre Martinetti

Christelle Gebara: Nonparabolic Γ-near innity operators In this talk we will first introduce the notion of NΓ-Hilbert space as introduced by Wolfgang Lück in 1997. Then, we will introduce the notion of Γ-Fredholm operators defined between N Γ-Hilbert space. And finally, the goal of the presentation is to define nonparabolic Γ-near infinity operators and show how they induces Γ-Fredholmness on admissible domains

Kai Mao: A going-down principle fo étale groupoids and Baum-Connes conjecture Using a going-down principle, Chabert, Echterhoff and Oyono-Oyono proved the injectivity of the assembly map for locally compact Hausdorff groups that are amenable at infinity. Bönicke proved the analogue for ample groupoids. In this talk we present the proof of the going-down principle for étale groupoids in general. It yields the injectivity of the assembly map for étale groupoids that are strongly amenable at infinity, which was firstly proved by Bönicke and Proietti via a categorical approach.

Yvann Gaudillot-Estrada: The quantum Mackey analogy by examples We present two distinct deformations of a given semisimple Lie group: the contraction onto the motion group, knwon as the Mackey-Higson deformation, and the quantisation of the symmetric pair related to a maximal compact subgroup, due to Letzter. For low dimensional examples, we explain how to assemble these two deformations into a continuous field of C*-algebras defined over a square, which has vertically constant K-theory. We also show how the representation theory varies along this field in these cases.

Quentin Karegar

Jean Renault: Sous-algèbres de Cartan dans les C*-algèbres. Dans mon premier exposé, je présenterai le cadre usuel de la théorie. Dans le deuxième exposé, je présenterai des développements plus récents concernant l’existence de sous-algèbres de Cartan dans les C*-algèbres nucléaires, les C*-algèbres de groupoides non effectifs et les C*-algèbres de groupoides non séparés.

Estelle Boffy: 20Positive and Contractive Projections on Schatten spaces A subspace of a Banach space E is said to be 1-complemented if it is the ra= nge of a contractive projection, and positively 1-complemented if the proje= ction is, in addition, positive. In the commutative Lp spaces, the descript= ion of contractive projections is well known (see, e.g., Ando's Theorem, 19= 66). In 1992, Arazy and Friedman provided a characterization of the 1-compl= emented subspaces of S^p(H), the noncommutative Lp space associated with B(= H), for p=E2=89=A02. In this talk, I will focus on the positively 1-complem= ented subspaces of S^p(H), and briefly on the case p=3D2

Michel Hilsum: Affine Kazdhan group actions on Schatten Ideals and Fixed point property. We shall review basics facts about group actions on Banach spaces by affine isometries and explain latest result concerning the case of Schatten ideals.

Patrick Poissel: A general pushforward theorem for compactly supported Fourier multipliers To any sufficiently regular distribution $m$ on a locally compact group is associated, by the mean of the Fourier transform, some sort of « differential operator with symbol $m$ » on the dual of this group which in general is merely a quantum group. In 1970, M. Jodeit Jr. has shown that if a compactly supported distribution on $\mathbf R^d$ is the symbol of a continuous linear operator from $L^p(\mathbf R^d)$ to $L^q(\mathbf R^d)$, then its pushforward by the canonical homomorphism from $\mathbf R^d$ to $\mathbf T^d$ is the symbol of a continuous linear operator from $\ell^p(\mathbf Z^d)$ to $\ell^q(\mathbf Z^d)$. We propose a generalisation of this result by characterising the continuous homomorphisms of locally compact groups by which, for all exponents $p$ and $q$, the pushforward of a compactly supported distribution symbol of a continuous linear operator from $L^p$ to $L^q$ is again the symbol of a continuous linear operator from $L^p$ to $L^q$ as being those which are open.

Eva-Maria Hekkelman: MOIs & toi With the framework of MOIs in mind from the first part of this mini-course, some arguments common in noncommutative geometry become recognisable as MOI identities. However, an adaptation is needed to deal with unbounded operators. I will discuss an abstract pseudodifferential calculus used in NCG and show a way of defining a functional calculus on this class of operators. This gives a straightforward path to define MOIs, and to obtain a variety of asymptotic expansions which will be familiar to the working noncommutative geometer.

Cedric Arhancet: Classical harmonic analysis viewed through the prism of noncommutative geometry This talk aims to connect noncommutative geometry with classical harmonic analysis on Banach spaces, with a particular emphasis on both classical and noncommutative Lp-spaces. The overarching goal is to show how the study of operators on Lp-spaces can be naturally integrated into the broader framework of noncommutative geometry, thereby opening new perspectives in analysis. https://arxiv.org/abs/2409.07750

Thibault Lescure: A noncommutative compactification of Bass-Serre Trees When a group G acts properly on a tree T, it is a classical result that the action of G on the Gromov compactification X of the tree is amenable, i.e. the crossed product C(X)xG is nuclear. We will introduce an analogue of the crossed product C*-algebra C(X)xG in the context of fundamental C*-algebras of graph of C*-algebras (Fima,Freslon 2013). This C*-algebra has interesting features which leads to new proofs of results concerning approximation properties and KK-theory.

Jean Renault

Yigang Qiu

Albert Nebout

Axel Saglio: Gaussian generating functionals on compact quantum groups One can define a quantum analogue of Levy processes on involutive bialgebras. These processes can be studied through the convolution semigroup of states formed by their marginal distributions, which in turn is fully characterized by its generating functional. Among those quantum Levy processes, we will focus on Gaussian processes and present the classification of their generating functionals (Gaussian generating functionals) on some well-known compact matrix quantum groups such as $S_n^+$, $U_n^+$, $O_n^+$ and the quantum automorphism group $\mathrm{Aut}^+(M_n,\mathrm{Tr})$.