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 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 $1/2$.

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.

This paper studies reversibility and transitivity of semigroups acting on homogeneous spaces. Properties of the reversor set of a subsemigroup acting on homogeneous spaces are presented. Let G be a topological group and L a subgroup of G. Assume that S is a subsemigroup of G with nonempty interior. It is presented a study of the reversibility of the S-action on $G/L$ in terms of the actions of S and L on homogeneous spaces of G. The results relate the reversibility and the transitivity of S in $G/L$ with the minimality of the action of L on homogeneous spaces of G. In addition, sufficient conditions for S to be right reversible in G if S is reversible in $G/L$ are presented.

We investigate when the algebraic numerical range is a C-spectral set in a Banach algebra. While providing several counterexamples based on classical ideas as well as combinatorial Banach spaces, we discuss positive results for matrix algebras and provide an absolute constant in the case of complex $2\times 2$-matrices with the induced $1$-norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.

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 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.

We investigate the structure of circle actions with the Rokhlin property, particularly in relation to equivariant $KK$-theory. Our main results are $\mathbb {T}$-equivariant versions of celebrated results of Kirchberg: any Rokhlin action on a separable, nuclear C*-algebra is $KK^{\mathbb {T}}$-equivalent to a Rokhlin action on a Kirchberg algebra; and two circle actions with the Rokhlin property on a Kirchberg algebra are conjugate if and only if they are $KK^{\mathbb {T}}$-equivalent.

In the presence of the Universal Coefficient Theorem (UCT), $KK^{\mathbb {T}}$-equivalence for Rokhlin actions reduces to isomorphism of a K-theoretical invariant, namely of a canonical pure extension naturally associated with any Rokhlin action, and we provide a complete description of the extensions that arise from actions on nuclear $C^*$-algebras. In contrast with the non-equivariant setting, we exhibit an example showing that an isomorphism between the $K^{\mathbb {T}}$-theories of Rokhlin actions on Kirchberg algebras does not necessarily lift to a $KK^{\mathbb {T}}$-equivalence; this is the first example of its kind, even in the absence of the Rokhlin property.

In this article, it is shown that the lattice of C$^*$-covers of an operator algebra does not contain enough information to distinguish operator algebras up to completely isometric isomorphism. In addition, four natural equivalences of the lattice of C$^*$-covers are developed and proven to be distinct. The lattice of C$^*$-covers of direct sums and tensor products are studied. Along the way key examples are found of operator algebras, each of which generates exactly n C$^*$-algebras up to $*$-isomorphism, and a simple operator algebra that is not similar to a C$^*$-algebra.

We introduce the concept of pseudo-trisections of smooth oriented compact 4-manifolds with boundary. The main feature of pseudo-trisections is that they have lower complexity than relative trisections for given 4-manifolds. We prove existence and uniqueness of pseudo-trisections, and further establish a one-to-one correspondence between pseudo-trisections and their diagrammatic representations. We next introduce the concept of pseudo-bridge trisections of neatly embedded surfaces in smooth oriented compact 4-manifolds. We develop a diagrammatic theory of pseudo-bridge trisections and provide examples of computations of invariants of neatly embedded surfaces in 4-manifolds using said diagrams.

A pebble tree is an ordered tree where each node receives some colored pebbles, in such a way that each unary node receives at least one pebble, and each subtree has either one more or as many leaves as pebbles of each color. We show that the contraction poset on pebble trees is isomorphic to the face poset of a convex polytope called pebble tree polytope. Beside providing intriguing generalizations of the classical permutahedra and associahedra, our motivation is that the faces of the pebble tree polytopes provide realizations as convex polytopes of all assocoipahedra constructed by K. Poirier and T. Tradler only as polytopal complexes.

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 analyze infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety X over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra controlling the deformation problem. As an application, we study infinitesimal deformations of the pairs given by a locally free sheaf and a subspace of its sections with a view toward Brill-Noether theory.

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.

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.

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 establish hyperweak boundedness of area functions, square functions, maximal operators, and Calderón–Zygmund operators on products of two stratified Lie groups.

We compute the class groups of full rank upper cluster algebras in terms of the exchange polynomials. This characterizes the UFDs among these algebras. Our results simultaneously generalize theorems of Garcia Elsener, Lampe, and Smertnig from 2019 and of Cao, Keller, and Qin from 2023. Furthermore, we show that every (upper) cluster algebra is a finite factorization domain.

In this article, we study a class of convective diffusive elliptic problem with Dirichlet boundary condition and measure data in variable exponent spaces. We begin by introducing an approximate problem via a truncation approach and Yosida’s regularization. Then, we apply the technique of maximal monotone operators in Banach spaces to obtain a sequence of approximate solutions. Finally, we pass to the limit and prove that this sequence of solutions converges to at least one weak or entropy solution of the original problem. Furthermore, under some additional assumptions on the convective diffusive term, we prove the uniqueness of the entropy solution.

We introduce the L-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integral weight in Kohnen plus space.