The Rise and Fall of Modern Plagues

Millennia separate smallpox, the oldest pandemic, and COVID-19, the newest. Both calamities arose from an opportunistic virus amid human crowding. A vaccine for smallpox existed since the late 18th century, but it took worldwide public health strategy to eradicate it. COVID-19 proceeds against a hive of scientific collaboration, but succeeds, so far, from weak containment policies. COVID-19 was first identified in China, but it rose to become the American pandemic.

Weak containment of measure-preserving group actions

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

Induction for locally compact quantum groups revisited

In this paper, we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

Weak Containment by Restrictions of Induced Representations

A QSIN group is a locally compact group G whose group algebra $\mathrm L^{1}(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if G is a QSIN group, H is a closed subgroup of G, and $\pi \!: H\to \mathcal B(\mathcal{H})$ is a unitary representation of H, then $\pi$ is weakly contained in $\Big (\mathrm{Ind}_{H}^{G}\pi \Big )|_{H}$. This provides a powerful tool in studying the C*-algebras of QSIN groups. In particular, it is shown that if G is a QSIN group which contains a copy of $\mathbb{F}_{2}$ as a closed subgroup, then $\mathrm C^{\ast }(G)$ is not locally reflexive and $\mathrm C^{\ast }_{r}(G)$ does not admit the local lifting property. Further applications are drawn to the “(weak) extendability” of Fourier spaces $\mathrm A_{\pi }$ and Fourier–Stieltjes spaces $\mathrm B_{\pi }$.

Dynamical properties of profinite actions

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.