On spectra of Koopman, groupoid and quasi-regular representations

We introduce unitary representations of continuous groupoids on continuous fields of Hilbert spaces. We investigate some properties of these objects and discuss some of the standard constructions from representation theory in this particular context. An important r\^{ole} is played by the regular representation. We conclude by discussing some operator algebra associated to continuous representations of groupoids; in particular, we analyse the relationship of continuous repres...

The aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand, how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(o...

For any countable group, and also for any locally compact second countable, compactly generated topological group, G, we show the existence of a "universal" hypercyclic (i.e. topologically transitive) representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of G.

We introduce the $C^*$-algebra $C^*(\kappa)$ generated by the Koopman representation $\kappa$ of an \'etale groupoid $G$ acting on a measure space $(X,\mu)$. We prove that for a level transitive self-similar action $(G,E)$ with $E$ finite and $|uE^1|$ constant, there is an invariant measure $\nu$ on $X=E^\infty$ and that $C^*(\kappa)$ is residually finite-dimensional with a normalized self-similar trace. We also discus $p$-fold similarities of Hilbert spaces in connection to ...

We show that weak containment of free ergodic measure-preserving actions of $\mathbf{F}_\infty$ is not equivalent to weak containment of the corresponding Koopman representations. This result is based on the construction of an invariant random subgroup of $\mathbf{F}_\infty$ which is supported on the maximal actions.

Let $G$ be a residually finite group. To any decreasing sequence $\mathcal S = (H_n)_n $ of finite index subgroups of $G$ is associated a unitary representation $\rho_{\mathcal S}$ of $G$ in the Hilbert space $\bigoplus_{n=0}^{+\infty} \ell^2 (G/H_n) $. This paper investigates the following question: when does the representation $\rho_{\mathcal S} $ weakly contain the regular representation $\lambda$ of $G$?

Let $S=\{p_1, \dots, p_r,\infty\}$ for prime integers $p_1, \dots, p_r.$ Let $X$ be an $S$-adic compact nilmanifold, equipped with the unique translation invariant probability measure $\mu.$ We characterize the countable groups $\Gamma$ of automorphisms of $X$ for which the Koopman representation $\kappa$ on $L^2(X,\mu)$ has a spectral gap. More specifically, we show that $\kappa$ does not have a spectral gap if and only if there exists a non-trivial $\Gamma$-invariant quotie...

We prove universal lower bounds for discrepancies (i.e. sizes of spectral gaps of averaging operators) of measure-preserving actions of a locally compact group on probability spaces. For example, a locally compact Hausdorff unimodular group $G$, acting continuously, by measure-preserving transformations, on a compact atomless probability space $(X,\nu)$, with an orbit $Gx_0$ of measure zero, contained in the support of $\nu$, and with compact stabilizer (i.e. $G_{x_0}$ is com...

Given a Hausdorff locally compact \'etale groupoid $\mathcal G$, we describe as a topological space the part of the primitive spectrum of $C^*(\mathcal G)$ obtained by inducing one-dimensional representations of amenable isotropy groups of $\mathcal G$. When $\mathcal G$ is amenable, second countable, with abelian isotropy groups, our result gives the description of $\operatorname{Prim} C^*(\mathcal G)$ conjectured by van Wyk and Williams. This, in principle, completely deter...

Actions of a locally compact group G on a measure space X give rise to unitary representations of G on Hilbert spaces. We review results on the rigidity of these actions from the spectral point of view, that is, results about the existence of a spectral gap for associated averaging operators and their consequences. We will deal both with spaces X with an infinite measure as well as with spaces with an invariant probability measure. The spectral gap property has several striki...