# Olivier GABRIEL

## Current projects

My current projects are the following:

• Construct and study twisted spectral triples constructed from ergodic actions of compact Lie groups on C*-algebras, in a joint effort with F. Fathizadeh.
• Analyse the permanence properties of spectral triples with R. Nest.

## Summary of research

My articles so far are organised around the topic of noncommutative principal bundles. I studied in particular noncommutative $U\left(1\right)$-principal bundles and their "smooth" versions with M. Grensing and more recently with F. Fathizadeh.

I have also investigated constructions of free actions of compact quantum groups, on my own and with M. Weber.

### Articles

• Pairings, K-theory and Cyclic Cohomology for Quantum Heisenberg Manifolds, Journal of Noncommutative Geometry 7 (2013), 499--524. Published version.

The Quantum Heisenberg Manifolds (QHMs) were introduced by Rieffel in 1989 as strict deformation quantizations of Heisenberg manifolds. They are examples of noncommutative $U\left(1\right)$-principal bundles. In this paper, we first determine their periodic cyclic homology and cohomology and then compute the Chern-Connes pairings between K-theory and cyclic cohomology.

• Six-Term Exact Sequences for Smooth Generalized Crossed Products, with M. Grensing, Journal of Noncommutative Geometry 7 (2013), 301--333. Published version.

We define smooth generalized crossed products and prove six-term exact sequences of Pimsner-Voiculescu type for periodic cyclic cohomology. This sequence may, in particular, be applied to smooth subalgebras of the quantum Heisenberg manifolds in order to compute the generators of their cyclic cohomology. Further, our results include the known results for smooth crossed products. Our proof is based on a combination of arguments from the setting of Cuntz-Pimsner algebras and the Toeplitz proof of Bott periodicity.

• Fixed points of compact quantum groups actions on Cuntz algebras, Annales Henri Poincaré 15 (2014), 1013--1036. Published version.

Given an action of a Compact Quantum Group (CQG) on a finite dimensional Hilbert space, we can construct an action on the associated Cuntz algebra. We study the fixed point algebra of this action, using classification theory à la Kirchberg-Phillips. Under certain conditions, we prove that the fixed point algebra is purely infinite and simple. We further identify it as a C*-algebra, compute its K-theory and prove a "stability property": the fixed points only depend on the CQG via its fusion rules. We apply the theory to $SU$q(N) and illustrate by explicit computations for $SU$q(2) and $SU$q(3). This construction provides examples of free actions of CQG (or "principal noncommutative bundles").

• On the Chern-Gauss-Bonnet theorem and conformally twisted spectral triples for C*-dynamical systems, with F. Fathizadeh, published version (in open access), Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 12 (2016), 016, 21 pages.

The analog of the Chern-Gauss-Bonnet theorem is studied for a $C*$-dynamical system consisting of a C*-algebra $A$ equipped with an ergodic action of a compact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$ is used to interpret the Chevalley-Eilenberg complex with coefficients in the natural smooth subalgebra of $A$ as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique $G$-invariant state on A, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on the smooth subalgebra and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

• Ergodic Actions and Spectral Triples, joint with M. Grensing, Journal of Operator Theory 76 (2016), 2, 307--335. Published version.

In this article, we give a general construction of spectral triples from certain Lie group actions on unital C*-algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes' axioms. This provides a link between the "algebraic" existence of ergodic action and the "analytic" finite summability property of the unbounded selfadjoint operator. More generally, for compact G we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illustrated by a host of examples - including noncommutative tori and quantum Heisenberg manifolds.

• Fixed Point Algebras for Easy Quantum Groups, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 12 (2016), 097, 21 pages, Published version.

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their $K$-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the $K$-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S+n, the free orthogonal quantum group $O+$n and the quantum reflection groups $Hs+$n. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.

### Preprints

• Generalized crossed products and spectral triples, ArXiv version, with M. Grensing.

We give a construction allowing to lift spectral triples to crossed products by Hilbert bimodules. The spectral triple one obtains is a concrete unbounded representative of the Kasparov product of the spectral triple and the Pimsner-Toeplitz extension associated to the crossed product by the Hilbert bimodule. To prove that the lifted spectral triple is the above-mentioned Kasparov product, we rely on operator-*-algebras and connexions.

• Norm inequality and partition of (pure) states in tensor product of smooth functions by matrices, pdf version.

Given a spectral triple, a noncommutative distance between states can be defined. Even for "compact" noncommutative spaces, infinite distances between states can arise. We first study this phenomenon in a general setting and then specialise to certain spectral triples over $C\left(M\right) \otimes M$n(C). The "bounded components" induce a partition of the space of states, which we describe in our particular case.

### In preparation

• Permanence properties of spectral triples, joint with R. Nest.

The goal of this research project is to investigate permanence properties for spectral triples: given a compact quantum group $G$ acting freely on a $C*$-algebra $A$ with fixed point algebra $B:=AG$ and a spectral triple on $B$, we want to construct a spectral triple on $A$ and describe its class in $K$-homology.

Last update: 9.12.2016 by Olivier Gabriel.