Entropy of measure-preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are those given by the techniques developed by Ollagnier and Pinchon on the one hand and the Ornstein–Weiss lemma on the other. We extend these two approaches to the context of actions of amenable topological groups. In contrast to the discrete setting, our results reveal a remarkable difference between the two concepts of entropy in the realm of non-discrete groups: while the first quantity collapses to 0 in the non-discrete case, the second yields a well-behaved invariant for amenable unimodular groups. Concerning the latter, we moreover study the corresponding notion of topological pressure, prove a Goodwyn-type theorem, and establish the equivalence with the uniform lattice approach (for locally compact groups admitting a uniform lattice). Our study elaborates on a version of the Ornstein–Weiss lemma due to Gromov.

If $\mu $ is a smooth measure supported on a real-analytic submanifold of ${\mathbb {R}}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu $ is a compact operator.

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.

We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.

We classify the irreducible unitary representations of closed simple groups of automorphisms of trees acting $2$-transitively on the boundary and whose local action at every vertex contains the alternating group. As an application, we confirm Claudio Nebbia’s CCR conjecture on trees for $(d_0,d_1)$-semi-regular trees such that $d_0,d_1\in \Theta $, where $\Theta $ is an asymptotically dense set of positive integers.

Let $G$ be a compact Abelian group and $E$ a subset of the group $\widehat {G}$ of continuous characters of $G$. We study Arens regularity-related properties of the ideals $L_E^1(G)$ of $L^1(G)$ that are made of functions whose Fourier transform is supported on $E\subseteq \widehat {G}$. Arens regularity of $L_E^1(G)$, the centre of $L_E^1(G)^{\ast \ast }$ and the size of $L_E^1(G)^\ast /\mathcal {WAP}(L_E^1(G))$ are studied. We establish general conditions for the regularity of $L_E^1(G)$ and deduce from them that $L_E^1(G)$ is not strongly Arens irregular if $E$ is a small-2 set (i.e. $\mu \ast \mu \in L^1(G)$ for every $\mu \in M_E^1(G)$), which is not a $\Lambda (1)$-set, and it is extremely non-Arens regular if $E$ is not a small-2 set. We deduce also that $L_E^1(G)$ is not Arens regular when $\widehat {G}\setminus E$ is a Lust-Piquard set.

Let $l\in \mathbb {N}_{\ge 1}$ and $\alpha : \mathbb {Z}^l\rightarrow \text {Aut}(\mathscr {N})$ be an action of $\mathbb {Z}^l$ by automorphisms on a compact nilmanifold $\mathscr{N}$. We assume the action of every $\alpha (z)$ is ergodic for $z\in \mathbb {Z}^l\smallsetminus \{0\}$ and show that $\alpha $ satisfies exponential n-mixing for any integer $n\geq 2$. This extends the results of Gorodnik and Spatzier [Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215 (2015), 127–159].

Let $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.

Let ${\mathcal A}$ be a Banach algebra, and let $\varphi $ be a nonzero character on ${\mathcal A}$. For a closed ideal I of ${\mathcal A}$ with $I\not \subseteq \ker \varphi $ such that I has a bounded approximate identity, we show that $\operatorname {WAP}(\mathcal {A})$, the space of weakly almost periodic functionals on ${\mathcal A}$, admits a right (left) invariant $\varphi $-mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$-mean. This generalizes a result due to Neufang for the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$, for a locally compact group G. Then we apply this result to the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$. Finally, we study the existence of left and right invariant $1$-means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$.

Let $\mathbb {V}$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, the pull–push form. In particular, it is always analytically dense when the pull–push form does not vanish. When the weight is two, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$th Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal {A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal {A}_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull–push form appears in this greater generality, we provide several tools to determine it, and we compute it in many examples.

In this paper, we address the problem of computing the topological entropy of a map $\psi : G \to G$, where G is a Lie group, given by some power $\psi (g) = g^k$, with k a positive integer. When G is abelian, $\psi $ is an endomorphism and its topological entropy is given by $h(\psi ) = \dim (T(G)) \log (k)$, where $T(G)$ is the maximal torus of G, as shown by Patrão [The topological entropy of endomorphisms of Lie groups. Israel J. Math. 234 (2019), 55–80]. However, when G is not abelian, $\psi $ is no longer an endomorphism and these previous results cannot be used. Still, $\psi $ has some interesting symmetries, for example, it commutes with the conjugations of G. In this paper, the structure theory of Lie groups is used to show that $h(\psi ) = \dim (T)\log (k)$, where T is a maximal torus of G, generalizing the formula in the abelian case. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.

Given an irreducible lattice $\Gamma $ in the product of higher rank simple Lie groups, we prove a co-finiteness result for the $\Gamma $-invariant von Neumann subalgebras of the group von Neumann algebra $\mathcal {L}(\Gamma )$, and for the $\Gamma $-invariant unital $C^*$-subalgebras of the reduced group $C^*$-algebra $C^*_{\mathrm {red}}(\Gamma )$. We use these results to show that: (i) every $\Gamma $-invariant von Neumann subalgebra of $\mathcal {L}(\Gamma )$ is generated by a normal subgroup; and (ii) given a weakly mixing unitary representation $\pi $ of $\Gamma $, every $\Gamma $-equivariant conditional expectation on $C^*_\pi (\Gamma )$ is the canonical conditional expectation onto the $C^*$-subalgebra generated by a normal subgroup.

Let G be a locally compact unimodular group, and let $\phi $ be some function of n variables on G. To such a $\phi $, one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the noncommutative $L_p$-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes $S_p(L_2(G))$. We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called “multiplicatively bounded $(p_1,\ldots ,p_n)$-norm” of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map $L_{p_1}(\mathbb {R}, S_{p_1}) \times L_{p_2}(\mathbb {R}, S_{p_2}) \rightarrow L_{1}(\mathbb {R}, S_{1})$, whenever $p_1$ and $p_2$ are such that $\frac {1}{p_1} + \frac {1}{p_2} = 1$. A similar result holds for certain Calderón–Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.

We extend the Burger–Mozes theory of closed, nondiscrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger–Mozes universal groups acting on the regular tree $T_{d}$ of degree $d\in \mathbb {N}_{\ge 3}$. Three applications are given. First, we characterize the automorphism types that the quasicentre of a nondiscrete subgroup of $\operatorname {\mathrm {Aut}}(T_{d})$ may feature in terms of the group’s local action. In doing so, we explicitly construct closed, nondiscrete, compactly generated subgroups of $\operatorname {\mathrm {Aut}}(T_{d})$ with nontrivial quasicentre, and see that the Burger–Mozes theory does not extend further to the transitive case. We then characterize the $(P_{k})$-closures of locally transitive subgroups of $\operatorname {\mathrm {Aut}}(T_{d})$ containing an involutive inversion, and thereby partially answer two questions by Banks et al. [‘Simple groups of automorphisms of trees determined by their actions on finite subtrees’, J. Group Theory 18(2) (2015), 235–261]. Finally, we offer a new view on the Weiss conjecture.

A connected, locally finite graph $\Gamma $ is a Cayley–Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on $\Gamma $ with compact, open vertex stabilizers. Define the minimal degree of G as the minimal degree of a Cayley–Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if $T_{d}$ denotes the d-regular tree, then the minimal degree of $\mathrm{Aut}(T_{d})$ is d for all $d\geq 2$.

In 2016, I solved a problem of de la Harpe from 2006: Is there a nondiscrete C$^{\ast }$-simple group? However the solution was not fully satisfactory, as the C$^{\ast }$-simple groups provided (and their operator algebras) are very close to discrete groups. All previously known examples are of this form. In this article I give yet another construction of nondiscrete C$^{\ast }$-simple groups. The statement in the title then follows. This in particular gives the first examples of nonelementary C$^{\ast }$-simple groups (in Wesolek’s sense).

For a locally compact metrisable group G, we study the action of ${\rm Aut}(G)$ on ${\rm Sub}_G$, the set of closed subgroups of G endowed with the Chabauty topology. Given an automorphism T of G, we relate the distality of the T-action on ${\rm Sub}_G$ with that of the T-action on G under a certain condition. If G is a connected Lie group, we characterise the distality of the T-action on ${\rm Sub}_G$ in terms of compactness of the closed subgroup generated by T in ${\rm Aut}(G)$ under certain conditions on the center of G or on T as follows: G has no compact central subgroup of positive dimension or T is unipotent or T is contained in the connected component of the identity in ${\rm Aut}(G)$. Moreover, we also show that a connected Lie group G acts distally on ${\rm Sub}_G$ if and only if G is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on ${\rm Sub}^a_G$, a subset of ${\rm Sub}_G$ consisting of closed abelian subgroups of G.

We prove that for a Banach algebra A having a bounded $\mathcal {Z}(A)$-approximate identity and for every $\mathbf {[IN]}$ group G with a weight w which is either constant on conjugacy classes or satisfies $w \geq 1$, $\mathcal {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) \otimes ^{\gamma } \mathcal {Z}(A)$. As an application, we discuss the conditions under which $\mathcal {Z}(L^{1}_{\omega }(G,A))$ enjoys certain Banach algebraic properties, such as weak amenability or semisimplicity.