The article deals with isometric dilation and commutant lifting for a class of n-tuples $(n\ge 3)$ of commuting contractions. We show that operator tuples in the class dilate to tuples of commuting isometries of BCL type. As a consequence of such an explicit dilation, we show that their von Neumann inequality holds on a one-dimensional variety of the closed unit polydisc. On the basis of such a dilation, we prove a commutant lifting theorem of Sarason’s type by establishing that every commutant can be lifted to the dilation space in a commuting and norm-preserving manner. This further leads us to find yet another class of n-tuples $(n\ge 3)$ of commuting contractions each of which possesses isometric dilation.

Let $\mathcal H$ be a complex, separable Hilbert space, and set . When $\dim \, \mathcal H$ is finite, we characterise the set and its norm-closure . In the infinite-dimensional setting, we characterise the intersection of with the set of biquasitriangular operators, and we exhibit an index obstruction to belonging to .

We expand the theory of 2-classifiers, that are a 2-categorical generalization of subobject classifiers introduced by Weber. The idea is to upgrade monomorphisms to discrete opfibrations. We prove that the conditions of 2-classifier can be checked just on a dense generator. The study of what is classified by a 2-classifier is similarly reduced to a study over the objects that form a dense generator. We then apply our results to the cases of prestacks and stacks, where we can thus look just at the representables. We produce a 2-classifier in prestacks that classifies all discrete opfibrations with small fibres. Finally, we restrict such 2-classifier to a 2-classifier in stacks. This is the main ingredient of a proof that Grothendieck 2-topoi are elementary 2-topoi. Our results also solve a problem posed by Hofmann and Streicher when attempting to lift Grothendieck universes to sheaves.

In this paper, we classify all global dynamics of the three-dimensional type-K monotone Lotka–Volterra system with the identical intrinsic growth rate inside the compactification of the positive octant of $\mathbb {R}^{3}$. By means of the replicator equations, it is proved that this system can have exactly $35$ topologically different phase portraits. As a consequence, we obtain the necessary and sufficient condition for the system to be bounded in the positive octant and verify that the limit set of any orbit of the compactified vector field associated with the system is an equilibrium.

Given a full right-Hilbert $\mathrm {C}^{*}$-module $\mathbf {X}$ over a $\mathrm {C}^{*}$-algebra A, the set $\mathbb {K}_{A}(\mathbf {X})$ of A-compact operators on $\mathbf {X}$ is the (up to isomorphism) unique $\mathrm {C}^{*}$-algebra that is strongly Morita equivalent to the coefficient algebra A via $\mathbf {X}$. As a bimodule, $\mathbb {K}_{A}(\mathbf {X})$ can also be thought of as the balanced tensor product $\mathbf {X}\otimes _{A} \mathbf {X}^{\mathrm {op}}$, and so the latter naturally becomes a $\mathrm {C}^{*}$-algebra. We generalize both of these facts to the world of Fell bundles over groupoids: Suppose $\mathscr {B}$ is a Fell bundle over a groupoid $\mathcal {H}$ and $\mathscr {M}$ is an upper semi-continuous Banach bundle over a principal $\mathcal {H}$-space X. If $\mathscr {M}$ carries a right-action of $\mathscr {B}$ and a sufficiently nice $\mathscr {B}$-valued inner product, then its imprimitivity Fell bundle $\mathbb {K}_{\mathscr {B}}(\mathscr {M})=\mathscr {M}\otimes _{\mathscr {B}} \mathscr {M}^{\mathrm {op}}$ is a Fell bundle over the imprimitivity groupoid of X, and it is the unique Fell bundle that is equivalent to $\mathscr {B}$ via $\mathscr {M}$. We show that $\mathbb {K}_{\mathscr {B}}(\mathscr {M})$ generalizes the “higher order” compact operators of Abadie–Ferraro in the case of saturated bundles over groups, and that the theorem recovers results such as Kumjian’s Stabilization trick.

We prove new fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras. In particular, we verify the transfer conjecture and the arithmetic transfer conjecture from Li and Mihatsch (2023, Preprint, arXiv:2307.11716) in cases of Hasse invariant $1/2$.

We prove that a C$^*$-algebra A has uniform property $\Gamma $ if the set of extremal tracial states, $\partial _e T(A)$, is a non-empty compact space of finite covering dimension and for each $\tau \in \partial _e T(A)$, the von Neumann algebra $\pi _\tau (A)"$ arising from the GNS representation has property $\Gamma $.

We prove that a group $\Gamma $ admits a discrete, topological (equivalently, smooth) action on some simply connected 3-manifold if and only if $\Gamma $ has a Cayley complex embeddable—with certain natural restrictions—in one of the following four 3-manifolds: (i) $\mathbb {S}^3$, (ii) $\mathbb {R}^3$, (iii) $\mathbb {S}^2 \times \mathbb R$, and (iv) the complement of a tame Cantor set in $\mathbb {S}^3$. The fact that these are the only simply connected 3-manifolds that allow such actions is a consequence of the Thurston–Perelman geometrization theorem.

We explore generalizations of the p-adic Simpson correspondence on smooth proper rigid spaces to principal bundles under rigid group varieties G. For commutative G, we prove that such a correspondence exists if and only if the Lie group logarithm is surjective. Second, we treat the case of general G on ordinary abelian varieties, in which case we prove a generalization of Faltings’ “small” correspondence to general rigid groups. On abeloid varieties, we also prove an analog of the classical Corlette–Simpson correspondence for principal bundles under linear algebraic groups.

In this paper, we prove an optimal isoperimetric inequality for spacelike, compact, star-shaped, and $2$-convex hypersurfaces in de Sitter space.