Continuous Homomorphism Research Articles

Homological properties of quotient divisible Abelian groups and compact groups dual to them

Homological properties of quotient divisible Abelian groups are studied. These groups form an important class of groups, which has been extensively studied in recent years. The first part of the paper is devoted to conditions for the triviality of extension groups in which one of the arguments is a quotient divisible group. Under certain additional assumptions, groups of homomorphisms from quotient divisible groups to reduced Abelian groups are described. Universality properties of quotient divisible Abelian groups are investigated. The second part of the paper considers homological properties of compact Abelian groups dual to quotient divisible groups in the sense of L.S. Pontryagin. Such groups are said to be “quotient toroidal.” Conditions for the triviality of group extensions in which one of the arguments is a quotient toroidal group are studied. Certain groups of continuous homomorphisms in which the second argument is a quotient toroidal group are described. The last part of the paper is devoted to conditions for the triviality of the groups of extensions of quotient divisible groups by compact quotient toroidal ones. The fundamental group of the topological space of a quotient toroidal group is characterized.

A quantitative generalization of Prodanov–Stoyanov Theorem on minimal Abelian topological groups

A topological group X is defined to have compact exponent if for some number n∈N the set {xn:x∈X} has compact closure in X. Any such number n will be called a compact exponent of X. Our principal result states that a complete Abelian topological group X has compact exponent (equal to n∈N) if and only if for any injective continuous homomorphism f:X→Y to a topological group Y and every y∈f(X)‾ there exists a positive number k (equal to n) such that yk∈f(X). This result has many interesting implications: (1) each minimal Abelian topological group is precompact (this is the famous Prodanov–Stoyanov Theorem); (2) an Abelian topological group is compact if and only if it is complete in each weaker Hausdorff group topology (this resolves a problem of Protasov and Zelenyuk of 2001); (3) an Abelian topological group X is complete and has compact exponent if and only if it is closed in each Hausdorff paratopological group containing X as a topological subgroup (this confirms a conjecture of Banakh and Ravsky of 2001).

On images of complete topologized subsemilattices in sequential semitopological semilattices

A topologized semilattice X is called complete if each non-empty chain Csubset X has inf Cin {bar{C}} and sup Cin {bar{C}}. We prove that for any continuous homomorphism h:Xrightarrow Y from a complete topologized semilattice X to a sequential Hausdorff semitopological semilattice Y the image h(X) is closed in Y.

The closedness of complete subsemilattices in functionally Hausdorff semitopological semilattices

A topologized semilattice X is complete if each non-empty chain C⊂X has inf⁡C∈C¯ and sup⁡C∈C¯. It is proved that for any complete subsemilattice X of a functionally Hausdorff semitopological semilattice Y the partial order ≤X={(x,y)∈X×X:xy=x} of X is closed in Y×Y and hence X is closed in Y. This implies that for any continuous homomorphism h:X→Y from a compete topologized semilattice X to a functionally Hausdorff semitopological semilattice Y the image h(X) is closed in Y. The functional Hausdorffness of Y in these two results can be replaced by the weaker separation axiom T→2δ, defined in this paper.

Injective tests of low complexity in the plane

AbstractWe study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not.

Topological tensor product of bimodules, complete Hopf algebroids and convolution algebras

Given a finitely generated and projective Lie–Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the appendices we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.

Algebraic monodromy groups of l-adic representations of Gal(ℚ∕ℚ)

In this paper we prove that for any connected reductive algebraic group G and a large enough prime $l$, there are continuous homomorphisms $$\mathrm{Gal}(\bar\mathbb Q/\mathbb Q) \to G(\bar\mathbb Q_l)$$ with Zariski-dense image, in particular we produce the first such examples for $SL_n, Sp_{2n}, Spin_n, E_6^{sc}$ and $E_7^{sc}$. To do this, we start with a mod-$l$ representation of $\mathrm{Gal}(\bar\mathbb Q/\mathbb Q)$ related to the Weyl group of $G$ and use a variation of Stefan Patrikis' generalization of a method of Ravi Ramakrishna to deform it to characteristic zero.

Abstract homomorphisms from locally compact groups to discrete groups

We show that every abstract homomorphism φ from a locally compact group L to a graph product GΓ, endowed with the discrete topology, is either continuous or φ(L) lies in a ‘small’ parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not ‘small’ is discrete. This extends earlier work by Morris-Nickolas.We also show the following. If L is a locally compact group and if G is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism φ:L→G is either continuous, or φ(L) is contained in the normalizer of a finite nontrivial subgroup of G. As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups.

Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids

We study factorization properties of continuous homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an ω-retractable submonoid of a product D = ∏ i ∈ I D i of topologized monoids and f : S → H is a continuous homomorphism to a topologized semigroup H with ψ ( H ) ≤ ω , then one can find a countable subset E of I and a continuous homomorphism g : p E ( S ) → H satisfying f = g ∘ p E ↾ S , where p E is the projection of D to ∏ i ∈ E D i . The same conclusion is valid if S contains the Σ -product Σ D ⊂ D . Furthermore, we show that in both cases, there exists the smallest by inclusion subset E ⊂ I with the aforementioned properties.

PROJECTIVE CLONE HOMOMORPHISMS

AbstractIt is known that a countable $\omega $ -categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone homomorphism. We investigate the relationship between the existence of a clone homomorphism to the projection clone, and the existence of such a homomorphism which is continuous and thus meets the above criterion.

(φ,ψ)- biprojective Banach algebras

The article studies the concept of a biprojective and pseudo amenable Banach algebra , where is a continuous homomorphism on and . We show if is contractible, then is biprojective. The converse holds, whenever is either unital or commutative and there exists such that .

Completeness and absolute H-closedness of topological semilattices

We find (completeness type) conditions on topological semilattices X,Y guaranteeing that each continuous homomorphism h:X→Y has closed image h(X) in Y.

Strict topologies and spectral measures

Let X be a completely regular Hausdorff space. Then the space of all bounded continuous complex functions on X, equipped with the natural strict topology β is a locally convex algebra with the jointly continuous multiplication. Let be a barreled quasicomplete locally convex Hausdorff space and denote the space of all continuous operators of into itself, equipped with the topology τs of simple convergence. We establish the integral representation (with respect to τs -Radon spectral measures -continuous unital algebra homomorphisms . In particular, if λ is a positive Radon measure on X and Lϱ is a Banach function space in we study a homomorphism given by the equality , where Mu (f ) = uf for f ∈ Lϱ.

Strong Markov processes and negative definite functions associated with non-Archimedean elliptic pseudo-differential operators

In this article we prove that the heat kernel attached to the non-Archimedean elliptic pseudodifferential operators determine a Feller semigroup and a uniformly stochastically continuous $$C_{0}$$-transition function of some strong Markov processes $${\mathfrak {X}}$$ with state space $${\mathbb {Q}}_{p}^{n}.$$ We explicitly write the Feller semigroup and the Markov transition function associated with the heat kernel. Also, we show that the symbols of these pseudo-differential operators are a negative definite function and moreover, that this symbols can be represented as a combination of a constant $$c\ge 0,$$ a continuous homomorphism $$l: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}$$ and a non-negative, continuous quadratic form $$q: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}.$$

Semigroups of continuous module homomorphisms on complex complete random normed modules*

In this paper, we are concerned with a bounded strongly continuous semigroup of continuous module homomorphisms on a complex complete random normed module. The main results of this section are a theorem, which establishes a differential equation associated with the bounded s.c. semigroup generalizing a central result in the paper [X. Zhang and M. Liu, On almost surely bounded semigroups of random linear operators, J. Math. Phys., 54:053517, 2013], and a theorem, which is a version of the Hille–Yosida theorem for the contraction semigroup of continuous module homomorphisms on complex complete RN module.

The Hoffman–Rossi Theorem for Operator Algebras

We study possible noncommutative (operator algebra) variants of the classical Hoffman–Rossi theorem from the theory of function algebras. In particular we give a condition on the range of a contractive weak* continuous homomorphism defined on an operator algebra A, which is necessary and sufficient (in the setting we explain) for a positive weak* continuous extension to any von Neumann algebra containing A.

Quotients of locally minimal groups

A topological group G is called locally minimal if there exists a neighbourhood V of 1 in G such that if H is a Hausdorff group and f:G→H is a continuous isomorphism such that f(V) is a neighbourhood of 1 in H, then f is open. This paper is focused on the study of quotients of locally minimal groups.A topological group G is called locally q-minimal if there exists a neighbourhood V of the identity of G such that whenever H is a Hausdorff group and f:G→H is a continuous surjective homomorphism such that f(V) is a neighbourhood of 1 in H, then f is open. We find a close connection between locally q-minimality and divisibility, by showing that a dense subgroup of Rn is locally q-minimal if and only if it is divisible. A description of the locally q-minimal subgroups of the n-dimensional torus Tn is also given.Two weaker versions of local q-minimality are proposed – a topological group G is:(a)locally t-minimal, if all Hausdorff quotients of G are locally minimal;(b)locallyq⁎-minimal, if there exists a neighbourhood V of 1 in G such that whenever H is a Hausdorff group and f:G→H is a continuous surjective homomorphism such that f(V) is a neighbourhood of 1 in H and ker⁡f⊆V, then f is open.While local q-minimality and local t-minimality are preserved by taking quotients, local q⁎-minimality fails to have this property (so does not coincide with local t-minimality), yet it has a relevant advantage. By means of an appropriate notion of local t-density, one can show that a dense subgroup H of a Hausdorff group G is locally q⁎-minimal if and only if G is locally q⁎-minimal and H is locally t-dense in G. Similar criterion for local q-minimality is not available (examples are given to show that a topological group with a dense locally q-minimal subgroup need not be locally q-minimal). The interrelations of these three versions of local minimality, as well as their prominence properties are studied in detail.

The interplay between weak topologies on topological semilattices

We study the interplay between three weak topologies on a topological semilattice X: the weak∘ topology wX∘ (generated by the base consisting of open subsemilattices of X), the weak• topology wX• (generated by the subbase consisting of complements to closed subsemilattices), and the I-weak topology wX (which is the weakest topology in which all continuous homomorphisms h:X→[0,1] remain continuous). Also we study the interplay between the weak topologies w•X, wX∘, wX of a topological semilattice X and some intrinsic topologies, determined by the order structure of the semilattice.We prove that the weak• topology wX• on a Hausdorff semitopological semilattice X is compact if and only if X is chain-compact in the sense that each closed chain in X is compact. For a compact Hausdorff topological semilattice X with topology τX we prove that τX=wX iff τX=wX• iff τX=wX∘.

The Kummerian property and maximal pro-p Galois groups

For a prime number p, we give a new restriction on pro-p groups G which are realizable as the maximal pro-p Galois group GF(p) for a field F containing a root of unity of order p. This restriction arises from Kummer Theory and the structure of the maximal p-radical extension of F. We study it in the abstract context of pro-p groups G with a continuous homomorphism θ:G→1+pZp, and characterize it cohomologically, and in terms of 1-cocycles on G. This is used to produce new examples of pro-p groups which do not occur as maximal pro-p Galois groups of fields as above.

Fission, fusion, and 6D RG flows

We show that all known 6D SCFTs can be obtained iteratively from an underlying set of UV progenitor theories through the processes of “fission” and “fusion”. Fission consists of a tensor branch deformation followed by a special class of Higgs branch deformations characterized by discrete and continuous homomorphisms into flavor symmetry algebras. Almost all 6D SCFTs can be realized as fission products. The remainder can be constructed via one step of fusion involving these fission products, whereby a single common flavor symmetry of decoupled 6D SCFTs is gauged and paired with a new tensor multiplet at the origin of moduli space, producing an RG flow “in reverse” to the UV. This leads to a streamlined labeling scheme for all known 6D SCFTs in terms of a few pieces of group theoretic data. The partial ordering of continuous homomorphisms mathfrak{s}mathfrak{u}(2)to {mathfrak{g}}_{mathrm{flav}} for {mathfrak{g}}_{mathrm{flav}} a flavor symmetry also points the way to a classification of 6D RG flows.