Continuous Homomorphism Research Articles

On Ψω-factorizable groups

A topological group G is called Ψω-factorizable (resp. M-factorizable) if every continuous real-valued function on G admits a factorization via a continuous homomorphism onto a topological group H with ψ(H)≤ω (resp. a first-countable group). The first purpose of this article is to discuss some characterizations of Ψω-factorizable groups. It is shown that a topological group G is Ψω-factorizable if and only if every continuous real-valued function on G is Gδ-uniformly continuous, if and only if for every cozero-set U of G, there exists a Gδ-subgroup N of G such that UN=U. Sufficient conditions on the Ψω-factorizable group G to be M-factorizable are that G is τ-fine and τ-steady for a cardinal τ.

All Hartman–Mycielski groups are minimally almost periodic

For each topological group G, it is shown that the Hartman–Mycielski group G• over G is minimally almost periodic, that is, all continuous homomorphisms of G• to compact Hausdorff topological groups are trivial. We also show that the group G•, over a compactly generated topological group G, is algebraically generated by a compact connected subset. Consequently, a wealth of minimally almost periodic, connected, compactly generated topological Abelian groups exist.

Big Mapping Class Groups and the Co-Hopfian Property

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

Symmetry of f-Vectors of Toric Arrangements in General Position and Some Applications

A toric hyperplane is the preimage of a point in a circle of a continuous surjective group homomorphism from the n-torus to the circle. A toric hyperplane arrangement is a finite collection of such hyperplanes. In this paper, we study the combinatorial properties of toric hyperplane arrangements on n-tori which are spanning and in general position. Specifically, we describe the symmetry of f-vectors arising in such arrangements and a few applications of the result to count configurations of hyperplanes.

Entire Monogenic Functions of Given Proximate Order and Continuous Homomorphisms

Infinite order differential operators appear in different fields of mathematics and physics. In the past decade they turned out to play a crucial role in the theory of superoscillations and provided new insight in the study of the evolution as initial data for the Schrödinger equation. Inspired by the infinite order differential operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. Precisely, we consider homomorphisms acting on functions in the kernel of the Dirac operator. For this class of functions, often called monogenic functions, we introduce the proximate order and prove some fundamental properties. As an important application, we are able to characterize infinite order differential operators that act continuously on spaces of monogenic entire functions.

Spectrum, homomorphisms and multipliers of Lau product of Banach algebras

Given Banach algebras $A$, $B$ and a continuous homomorphism $\theta:B\longrightarrow A$ with $\Vert\theta\Vert \leq1$, we obtain characterization of spectrum, homomorphisms and multipliers of $A\times_{\theta}B$, which is a strongly splitting Banach algebra extension of $B$ by $A$. Also we characterize the semisimplicity of these algebras.

Unitary groups, $\boldsymbol{K}$ -theory, and traces

AbstractWe show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$ -theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).

Duality and stable compactness in Orlicz-type modules

Orlicz-type modules are module analogues of classical Orlicz spaces. We study duality and stable compactness in Orlicz-type modules. We characterize the conditional Köthe dual of an Orlicz-type module as the space of all σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}-order continuous module homomorphisms. We find an order continuity criterion for stable compactness in Orlicz-type modules. As an application, we obtain a robust representation result for conditional risk measures on Orlicz spaces.

Homomorphisms of L1 algebras and Fourier algebras

We investigate conditions for the extendibility of continuous algebra homomorphisms ϕ from the Fourier algebra A(F) of a locally compact group F to the Fourier-Stieltjes algebra B(G) of a locally compact group G to maps between the corresponding L∞ algebras which are weak* continuous. When ϕ is completely bounded and F is amenable, it is induced by a piecewise affine map α:Y→F where Y⊆G. We show that extendibility of ϕ is equivalent to α being an open map. We also study the dual problem for contractive homomorphisms ϕ:L1(F)→M(G). We show that ϕ induces a w* continuous homomorphism between the von Neumann algebras of the groups if and only if the naturally associated map θ (Greenleaf [1965], Stokke [2011]) is a proper map.

Real-valued measurable cardinals and sequentially continuous homomorphisms

A. V. Arkhangel'skiĭ asked in 1981 if the variety V of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety V is a proper subclass of the class of all topological groups. A topological group G is called g-sequential if for any topological group H any sequentially continuous homomorphism G→H is continuous. We introduce the concept of a g-sequential cardinal and prove that a locally compact group is g-sequential if and only if its local weight is not a g-sequential cardinal. The product of a family of non-trivial g-sequential topological groups is g-sequential if and only if the cardinal of this family is not g-sequential. Suppose G is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then G is g-sequential if and only if its weight is not a g-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.

Homomorphic images of algebraic groups

We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.

Krull dimension of the negligible quotient in mod p cohomology of a finite group

For a finite group G, a G-module M, and a field F, an element u∈Hd(G,M) is negligible over F if for each field extension L/F and every continuous group homomorphism from Gal(Lsep/L) to G, u belongs to the kernel of the induced homomorphism Hd(G,M)→Hd(L,M). For p a prime and a trivial G-action on the coefficients, the negligible elements in the cohomology ring H⁎(G,Z/pZ) form an ideal. We compute the generators of the negligible ideal in the mod p cohomology of elementary abelian p-groups. We further show that when p is odd or p=2 and either |G| is odd or F is not formally real, the Krull dimension of the quotient of mod p cohomology by the negligible ideal is 0. However, when p=2, |G| is even, and F is formally real, the Krull dimension of the quotient of mod 2 cohomology of a finite 2-group by the negligible ideal is 1.

On the strict topology of the multipliers of a JB^*-algebra

We introduce the Jordan-strict topology on the multiplier algebra of a JB^*-algebra, a notion which was missing despite the forty years passed after the first studies on Jordan multipliers. In case that a C^*-algebra A is regarded as a JB^*-algebra, the J-strict topology of M(A) is precisely the well-studied C^*-strict topology. We prove that every JB^*-algebra {mathfrak {A}} is J-strict dense in its multiplier algebra M({mathfrak {A}}), and that latter algebra is J-strict complete. We show that continuous surjective Jordan homomorphisms, triple homomorphisms, and orthogonality preserving operators between JB^*-algebras admit J-strict continuous extensions to the corresponding type of operators between the multiplier algebras. We characterize J-strict continuous functionals on the multiplier algebra of a JB^*-algebra {mathfrak {A}}, and we establish that the dual of M({mathfrak {A}}) with respect to the J-strict topology is isometrically isomorphic to {mathfrak {A}}^*. We also present a first application of the J-strict topology of the multiplier algebra, by showing that under the extra hypothesis that {mathfrak {A}} and {mathfrak {B}} are sigma -unital JB^*-algebras, every surjective Jordan ^*-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from {mathfrak {A}} onto {mathfrak {B}} admits an extension to a surjective J-strict continuous Jordan ^*-homomorphism (respectively, triple homomorphism or continuous orthogonality preserving operator) from M({mathfrak {A}}) onto M({mathfrak {B}}).

On bM-ω-balancedness and [formula omitted]-factorizability of para(semi)topological groups

In this article, we introduce a notion which is called bM-ω-balancedness for semitopological groups. We show that a semitopological group G is topologically isomorphic to a subgroup of the product of a family of metrizable semitopological groups if and only if G is T0bM-ω-balanced. Thus a semitopological group G is T0bM-ω-balanced if and only if G is regular, has property (⁎) and countable index of regularity. The notion of property (⁎) is in the sense of I. Sánchez in [16].We show that if G is a T0bM-ω-balanced weakly Lindelöf paratopological group with a q-point, then for every continuous real-valued function f:G→R, there exist a continuous clopen homomorphism π:G→H onto a separable metrizable paratopological group H such that ker(π) is countably compact and a continuous real-valued function g:H→R such that f=g∘π.We introduce the notions of M (DE)-factorizability for semitopological groups. We show that a semitopological group G is Tychonoff and M-factorizable if and only if G is T0bM-ω-balanced and has property ω-QU. If G is a Tychonoff M-factorizable semitopological group and f:G→H is an open continuous homomorphism onto a semitopological group H such that ker(f) is countably compact, then H is DE-factorizable.

Groups of convex bodies

In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen’s polytope algebra, with the universal property that continuous valuations on convex bodies correspond to continuous homomorphisms on the group of convex bodies. To study this group, we first obtain a version of McMullen polynomiality for valuations that take values not in fields or vector spaces, but in abelian groups. Using this, we are able to equip the group of convex bodies with a grading that consists of real vector spaces in all positive degrees, mirroring one of the main structural properties of the polytope algebra. It is hoped that this work can serve as the starting point for a K-theoretic interpretation of valuations on convex bodies.

ON EQUIVALENCE RELATIONS INDUCED BY LOCALLY COMPACT ABELIAN POLISH GROUPS

AbstractGiven a Polish groupG, let$E(G)$be the right coset equivalence relation$G^{\omega }/c(G)$, where$c(G)$is the group of all convergent sequences inG. The connected component of the identity of a Polish groupGis denoted by$G_0$.Let$G,H$be locally compact abelian Polish groups. If$E(G)\leq _B E(H)$, then there is a continuous homomorphism$S:G_0\rightarrow H_0$such that$\ker (S)$is non-archimedean. The converse is also true whenGis connected and compact.For$n\in {\mathbb {N}}^+$, the partially ordered set$P(\omega )/\mbox {Fin}$can be embedded into Borel equivalence relations between$E({\mathbb {R}}^n)$and$E({\mathbb {T}}^n)$.

On topological M-injective modules

Let R be a topological ring and M be a topological R-module. A topological R-module U is called a topological M-injective if for every continuous monomorphism f:K→M, where K is an open submodule of M, and for every continuous homomorphism g:K→U, there exists a continuous homomorphism h:M→U such that hf=g. A topological M-injective module is a generalization of a topological injective module defined by Goldman and Sah [14]. An infinite direct sum of injective modules is not necessary injective. In this paper, the properties of topological M-injective modules are investigated. We prove that an infinite direct sum of topological M-injective modules is also topological M-injective if the direct sum is an open submodule of the direct product of topological M-injective modules.

The structure of KMS weights on étale groupoid C*-algebras

We generalise a number of classical results from the theory of KMS states to KMS weights in the setting of C^{*} -dynamical systems arising from a continuous groupoid homomorphism c:\mathcal{G} \to \mathbb{R} on a locally compact second countable Hausdorff étale groupoid \mathcal{G} . In particular, we generalise Neshveyev's Theorem to KMS weights.

The fineness index of topological groups and [formula omitted]-factorizable groups

A topological group G is called M-factorizable (resp. R-factorizable) if every continuous real-valued function on G admits a factorization via a continuous homomorphism onto a first-countable (resp. second-countable) group. The purpose of this paper is a further study of M-factorizability in topological groups. It is shown that a topological group G is M-factorizable if and only if G has property ω-U, i.e., every continuous real-valued function on G is ω-uniform continuous. To achieve this goal, we introduce a new cardinal function of topological groups, which is called the fineness index. The cardinal invariance is also used to characterize Mm-factorizable groups which are generalization of m-factorizable groups introduced in [2].

Continuous homomorphic images to point-separating classes of topological groups

A topological group G is said to be a (C→D)-group if each continuous homomorphic image of G to an arbitrary group contained in the class C is automatically contained in D as well. When C is the class of compact groups, and D is the singleton class of the trivial group, the corresponding class of (C→D)-groups coincides with the traditional minimally almost periodic (MinAP) groups of von Neumann. In this paper we characterize classes C of topological groups for which the corresponding (C→D)-groups can be described in the following way when D is closed under continuous homomorphisms: G is a (C→D)-group if and only if all topological group quotients of G contained in C are automatically contained in D. As an application, we characterize the algebraic structure of Abelian groups which admit a group topology where all continuous homomorphic images to totally disconnected groups are all algebraically torsion groups, or bounded groups.