Compact Open Subset Research Articles

Square-integrable representations and the coadjoint action of solvable Lie groups

Abstract We characterize the square-integrable representations of (connected, simply connected) solvable Lie groups in terms of the generalized orbits of the coadjoint action. We prove that the normal representations corresponding, via the Pukánszky correspondence, to open coadjoint orbits are type I, not necessarily square-integrable representations. We show that the quasi-equivalence classes of type I square-integrable representations are in bijection with the simply connected open coadjoint orbits, and the existence of an open coadjoint orbit guarantees the existence of a compact open subset of the space of primitive ideals of the group. When the nilradical has codimension 1, we prove that the isolated points of the primitive ideal space are always of type I. This is not always true for codimension greater than 2, as shown by specific examples of solvable Lie groups that have dense, but not locally closed, coadjoint orbits.

The spectrum problem for Abelian $$\ell $$-groups and MV-algebras

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian $$\ell $$ -groups. As a first main result, we show that a topological space X is the prime spectrum of an MV-algebra if and only if X is spectral, and the lattice K(X) of compact open subsets of X is a closed epimorphic image of the lattice of “cylinder rational polyhedra” (a natural generalization of rational polyhedra) of $$[0,1]^Y$$ for some set Y. As a second main result we extend our results to Abelian $$\ell $$ -groups. That is, a topological space X is the prime spectrum of an Abelian $$\ell $$ -group if and only if X is generalized spectral, and the lattice K(X) is a closed epimorphic image of the lattice of “cylinder rational cones” (a generalization of rational cones) in $${{\mathbb {R}}}^Y$$ for some set Y. Finally, we axiomatize, in monadic second order logic, the Belluce lattices of free MV-algebras (equivalently, the lattice of cylinder rational polyhedra) of dimension 1, 2 and infinite, and we study the problem of describing Belluce lattices in certain fragments of second order logic.

Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces

Let X be a locally compact zero-dimensional space, let S be an equicontinuous set of homeomorphisms such that 1 \in S = S^{-1} , and suppose that \overline{Gx} is compact for each x \in X , where G = \langle S \rangle . We show in this setting that a number of conditions are equivalent: (a) G acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every compact open subset U of X , there is F \subseteq G finite such that \bigcap_{g \in F}g(U) is G -invariant. All of these are equivalent to a notion of recurrence, which is a variation on a concept of Auslander–Glasner–Weiss. It follows in particular that the action is distal if and only if it is equicontinuous.

Dense subalgebras of purely infinite simple groupoidC*-algebras

AbstractA simple Steinberg algebra associated to an ample Hausdorff groupoidGis algebraically purely infinite if and only if the characteristic functions of compact open subsets of the unit space are infinite idempotents. If a simple Steinberg algebra is algebraically purely infinite, then the reduced groupoid$C^*$-algebra$C^*_r(G)$is simple and purely infinite. But the Steinberg algebra seems too small for the converse to hold. For this purpose we introduce an intermediate *-algebraB(G) constructed using corners$1_U C^*_r(G) 1_U$for all compact open subsetsUof the unit space of the groupoid. We then show that ifGis minimal and effective, thenB(G) is algebraically properly infinite if and only if$C^*_r(G)$is purely infinite simple. We apply our results to the algebras of higher-rank graphs.

Using the Steinberg algebra model to determine the center of any Leavitt path algebra

Given an arbitrary graph, we describe the center of its Leavitt path algebra over a commutative unital ring. Our proof uses the Steinberg algebra model of the Leavitt path algebra. A key ingredient is a characterization of compact open invariant subsets of the unit space of the graph groupoid in terms of the underlying graph: an open invariant subset is compact if and only if its associated hereditary and saturated set of vertices satisfies Condition (F). We also give a basis of the center. Its cardinality depends on the number of minimal compact open invariant subsets of the unit space.

Ideal structure and pure infiniteness of ample groupoid -algebras

In this paper, we study the ideal structure of reduced$C^{\ast }$-algebras$C_{r}^{\ast }(G)$associated to étale groupoids$G$. In particular, we characterize when there is a one-to-one correspondence between the closed, two-sided ideals in$C_{r}^{\ast }(G)$and the open invariant subsets of the unit space$G^{(0)}$of$G$. As a consequence, we show that if$G$is an inner exact, essentially principal, ample groupoid, then$C_{r}^{\ast }(G)$is (strongly) purely infinite if and only if every non-zero projection in$C_{0}(G^{(0)})$is properly infinite in$C_{r}^{\ast }(G)$. We also establish a sufficient condition on the ample groupoid$G$that ensures pure infiniteness of$C_{r}^{\ast }(G)$in terms of paradoxicality of compact open subsets of the unit space$G^{(0)}$. Finally, we introduce the type semigroup for ample groupoids and also obtain a dichotomy result: let$G$be an ample groupoid with compact unit space which is minimal and topologically principal. If the type semigroup is almost unperforated, then$C_{r}^{\ast }(G)$is a simple$C^{\ast }$-algebra which is either stably finite or strongly purely infinite.

Hearing pseudoconvexity in Lipschitz domains with holes via $${\bar{\partial }}$$ ∂ ¯

Let \(\Omega ={\widetilde{\Omega }}{\setminus } \overline{D}\) where \({\widetilde{\Omega }}\) is a bounded domain with connected complement in \({\mathbb {C}}^n\) (or more generally in a Stein manifold) and D is relatively compact open subset of \({\widetilde{\Omega }}\) with connected complement in \(\widetilde{\Omega }\). We obtain characterizations of pseudoconvexity of \({\widetilde{\Omega }}\) and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups of \(\Omega \) on various function spaces. In particular, we show that if the boundaries of \({\widetilde{\Omega }}\) and D are Lipschitz and \(C^2\)-smooth respectively, then both \({\widetilde{\Omega }}\) and D are pseudoconvex if and only if 0 is not in the spectrum of the \(\overline{\partial }\)-Neumann Laplacian of \(\Omega \) on (0, q)-forms for \(1\le q\le n-2\) when \(n\ge 3\); or 0 is not a limit point of the spectrum of the \(\overline{\partial }\)-Neumann Laplacian on (0, 1)-forms when \(n=2\).

Continuous ω q -Plurisubharmonic Exhaustion Functions on Kähler Manifolds

In this paper, we study continuous ωq-plurisubharmonic exhaustion functions. It is shown that if (M,ω) is a Kahler manifold and Ω⋐M is a relatively compact open subset such that there exists a smooth strictly plurisubharmonic function on M and for every a∈∂Ω, there exists a neighborhood Ua of a in M such that Ua∩Ω has a continuous ωq-plurisubharmonic exhaustion function then Ω has a continuous ωq-plurisubharmonic exhaustion function.

Automorphisms and connections on Higgs bundles over compact Kähler manifolds

Let (E,φ) be a Higgs vector bundle over a compact connected Kähler manifold X. Fix any filtration of E by coherent analytic subsheaves in which each sheaf is preserved by the Higgs field, and each successive quotient is a torsionfree and stable Higgs sheaf. Denote by G the direct sum of these stable quotients, and let the singular set of G be called S⊂X. We construct a 1-parameter family of filtration preserving C∞ isomorphisms Φt:G|X∖S→E|X∖S, and a Hermitian metric h on G|X\\S, such that as t→+∞, the Chern connection for the Hermitian Higgs bundle (Φt⁎E,Φt⁎φ,h)|X\\S converges, in the C∞ Fréchet topology over any relatively compact open subset of X\\S, to the direct sum of the Yang–Mills–Higgs connections on the direct summands in G.We also prove an analogous result for principal Higgs G-bundles on X.

Grauert's theorem for subanalytic open sets in real analytic manifolds

By an open neighbourhood in $\mathbb{C}^{n}$ of an open subset $\varOmega$ of $\mathbb{R}^n$ we mean an open subset $\varOmega'$ of $\mathbb{C}^n$ such that $\mathbb{R}^n\cap\varOmega'=\varOmega.$ A well known result of H. Grauert implies that any

On measure-preserving $\mathcal {C}^1$ transformations of compact-open subsets of non-archimedean local fields

We introduce the notion of a locally scaling transformation defined on a compactopen subset of a non-archimedean local field. We show that this class encompasses the Haar measure-preserving transformations defined by C (in particular, polynomial) maps, and prove a structure theorem for locally scaling transformations. We use the theory of polynomial approximation on compact-open subsets of non-archimedean local fields to demonstrate the existence of ergodic Markov, and mixing Markov transformations defined by such polynomial maps. We also give simple sufficient conditions on the Mahler expansion of a continuous map Zp → Zp for it to define a Bernoulli transformation.

Finitary formal topologies and Stone’s representation theorem

We study the concept of finitary formal topology, a point-free version of a topological space with a basis of compact open subsets. The notion of finitary formal topology is defined from the perspective of the Basic Picture (introduced by the second author) and thus it is endowed with a binary positivity relation. As an application, we prove a constructive version of Stone’s representation theorem for distributive lattices. We work within the framework of a minimalist foundation (as proposed by Maria Emilia Maietti and the second author). Both inductive and co-inductive methods are used in most proofs.

Hida families and rational points on elliptic curves

Let E be an elliptic curve over Q of conductor N = Mp, having a prime p‖N of multiplicative reduction. Because E is modular, it corresponds to a normalised weight two eigenform on Γ0(N), whose q-expansion is denoted f = ∑n anq . LetX := hom(Z×p ,Z×p ) Z/(p−1)Z×Zp, which contains Z as a dense subset by associating to k ∈ Z the character x → xk−2. Denote by A(U) the ring of Cp-valued p-adic analytic functions on a compact open subset U of X. Hida’s theory associates to f a neighborhood U of 2 ∈ X (which can be assumed, for simplicity, to be contained in the residue class of 2 modulo p − 1) and a formal q-expansion

Exposed sets in potential theory

Let U be a relatively compact open subset of a harmonic space, and H ( U ) be the function space of all continuous functions on U ¯ which are harmonic on U. We give a complete characterization of the H ( U ) -exposed subsets of U ¯ . This extends the results of [J. Lukeš, T. Mocek, M. Smrčka, J. Spurný, Choquet like sets in function spaces, Bull. Sci. Math. 127 (2003) 397–437].

Continuity of eigenvalues for Schrödinger operators, $L^p$ -properties of Kato type integral operators

Given an arbitrary relatively compact (finely) open subset of ${\mathbb R}^d, \mu$ -eigenvalues of $-A(D)+\nu$ are studied where $A(D)$ is the Dirichlet Laplacian on D and $\mu,\nu$ are measures on ${\mathbb R}^d$ such that $G_X^\mu$ is continuous and $G_X^\nu$ is bounded for every ball X in ${\mathbb R}^d (G_X$ being Green's function for X). Moreover, it is shown that these eigenvalues depend continuously on D and $\nu$ . The results are based on very general compactness and convergence properties of integral operators of Kato type which are developed before.

Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces

We prove the compactness of the imbedding of the Sobolev space $W^{1,2}_0(\Omega)$ into $L^2(\Omega)$ for any relatively compact open subset $\Omega$ of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Holder continuous heat kernel.

On domains witnessing increase in information

<p>The paper considers algebraic directed-complete partial orders with a semi-regular Scott topology, called regular domains. As is well know, the category of Scott domains and continuous maps is Cartesian closed. This is no longer true, if the domains are required to be regular. Two Cartesian closed subcategories of the regular Scott domains are exhibited: regular dI-domains with stable maps and strongly regular Scott domains with continuous maps. Here a Scott domains is strongly regular if all of its compact open subsets are regular open. In one considers only embeddings of dependent products and sums. Moreover, they are w-cocomplete and their object classes are closed under several constructions used in programming language semantics. It follows that recursive domains equations can be solved and models of typed and untyped lambda calculi can be constructed. Both kinds of domains can be udes in giving meaning to programming language constructs.</p>

Connectifications of metrizable spaces

We answer a question of Alas, Tkačenko, Tkachuk, and Wilson by constructing a metrizable space with no compact open subsets which cannot be densely embedded in a connected metrizable (or even perfectly normal) space. We also obtain a result that implies that every nowhere locally compact metrizable space can be densely embedded in a connected metrizable space.

Fonctions méromorphes sur un ouvert localement pseudoconvexe étalé au-dessus d'une variété projective

Let II : U → V be a complex space spread over an algebraic projective manifold V of dimension n, U being locally pseudoconvex in the sense of Hartogs. We show that we can immerse U in. Morever, if W is a relatively compact open subset in U, which is pseudoconvex (in the sense of Hartogs), then every meromorphic function on U is, on W, a quotient of two holomorphic sections of the bundle II * (()(l)).

Subspaces of connected spaces

A connectification of a topological space X is a connected Hausdorff space that contains X as a dense subspace. Watson and Wilson have noted that a Hausdorff space with a connectification has no nonempty proper clopen H-closed subspaces. Here it is proven that a Hausdorff space in which every nonempty proper clopen set is not feebly compact and the cardinality of the set of clopen sets is at most 2 c is connectifiable. This result is used to show that every metric space with no nonempty proper clopen H-closed subspace is connectifiable, answering a question asked by Watson and Wilson. Also, there is a nonconnectifiable, Hausdorff space of cardinality c with no proper H-closed subspace. Using the set-theoretic hypothesis p = c, an example of a nonconnectifiable, normal Hausdorff space of cardinality c is constructed which has no nonempty compact open subset. This space is locally compact at all but one point, and if the continuum hypothesis is assumed it is first countable. This space provides a solution to questions asked by Watson and Wilson as well as Mack. The paper concludes by examining when extremally disconnected Tychonoff spaces have Tychonoff connectifications.