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Abstract We extract a precise internal description of the sequential commutation equivalence relation introduced in [14] for tracial von Neumann algebras. As an application we, prove that if a tracial von Neumann algebra $$N$$ is generated by unitaries $$\{u_{i}\}_{i\in \mathbb{N}}$$ such that $$u_{i}\sim u_{j}$$ (i.e., there exists a finite set of Haar unitaries $$\{w_{i}\}_{i=1}^{n}$$ in $$N^{\mathcal{U}}$$ such that $$[u_{i}, w_{1}]= [w_{k}, w_{k+1}]=[w_{n},u_{j}]=0$$ for all $$1\leq k< n$$), then $$N$$ is singly generated. This generalizes and recovers several known single generation phenomena for II$$_{1}$$ factors in the literature with a unified proof. 

The emergence and mutation of pathogenic viruses have been occurring at an unprecedented rate in recent decades. The coronavirus disease 2019 (COVID-19) pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has developed into a global public health crisis due to extensive viral transmission. In situ RNA mapping has revealed angiotensin-converting enzyme 2 (ACE2) expression to be highest in the nose and lower in the lung, pointing to nasal susceptibility as a predominant route for infection and the cause of subsequent pulmonary effects. By blocking viral attachment and entry at the nasal airway using a cyclodextrin-based formulation, a preventative therapy can be developed to reduce viral infection at the site of entry. Here, we assess the safety and antiviral efficacy of cyclodextrin-based formulations. From these studies, hydroxypropyl beta-cyclodextrin (HPBCD) and hydroxypropyl gamma-cyclodextrin (HPGCD) were then further evaluated for antiviral effects using SARS-CoV-2 pseudotypes. Efficacy findings were confirmed with SARS-CoV-2 Delta variant infection of Calu-3 cells and using a K18-hACE2 murine model. Intranasal pre-treatment with HPBCD-based formulations reduced viral load and inflammatory signaling in the lung. In vitro efficacy studies were further conducted using lentiviruses, murine hepatitis virus (MHV), and influenza A virus subtype H1N1. These findings suggest HPBCD may be used as an agnostic barrier against transmissible pathogens, including but not limited to SARS-CoV-2. 

Human activities are accelerating rates of biological invasions and climate-driven range expansions globally, yet we understand little of how genomic processes facilitate the invasion process. Although most of the literature has focused on underlying phenotypic correlates of invasiveness, advances in genomic technologies are showing a strong link between genomic variation and invasion success. Here, we consider the ability of genomic tools and technologies to (i) inform mechanistic understanding of biological invasions and (ii) solve real-world issues in predicting and managing biological invasions. For both, we examine the current state of the field and discuss how genomics can be leveraged in the future. In addition, we make recommendations pertinent to broader research issues, such as data sovereignty, metadata standards, collaboration, and science communication best practices that will require concerted efforts from the global invasion genomics community.