Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let C be an Abelian group. An Abelian group A in some class of Abelian groups is said to be _C H-definable in the class if, for any group B\in , it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _C H-definable in , then the class is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.     

On definability of a periodic EndE+-group by its endomorphism group

Let A be a class of Abelian groups, A ∈ A, and End(A) be the additive endomorphism group of the group A. The group A is said to be defined by its endomorphism group in the class {ie208-01} if for every group B ∈ B such that End(B) ≅ End(A) the isomorphism B ≅ A holds. The paper considers the problem of definability of a periodic Abelian group A such that End-End(A) ≅ End(A). The classes of periodical Abelian groups, of divisible Abelian groups, of reduced Abelian groups, of nonreduced Abelian groups, and of all Abelian groups are investigated in this paper. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 123–131, 2007.     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.     

On a Problem Related to Homomorphism Groups in the Theory of Abelian Groups

In this paper, for any reduced Abelian group A whose torsion-free rank is infinite, we construct a countable set A(A) of Abelian groups connected with the group A in a definite way and such that for any two different groups B and C from the set A(A) the groups B and C are isomorphic but Hom(B,X) ? Hom(C,X) for any Abelian group X. The construction of such a set of Abelian groups is closely connected with Problem 34 from L. Fuchs’ book “Infinite Abelian Groups,” Vol. 1.     

A characterization of ℵ1-free abelian groups and its application to the chase radical

A groupA is an ℵ₁-free abelian group iffA is a subgroup of the Boolean power Z^(B) for some complete Boolean algebraB. The Chase radicalvA=Σ{C≦A: Hom(C, Z)=0 &C is countable). The torsion class {A:vA=A} is not closed under uncountable direct products.     

Rings of quasi-endomorphisms of some direct sums of torsion-free Abelian groups

We consider a representation of quasi-endomorphisms of Abelian torsion-free groups of rank 4 bymatrices of order 4 over the field of rational numbers Q. We obtain a classification for quasi-endomorphism rings of Abelian torsion-free groups of rank 4 quasi-decomposable into a direct sum of groups A ₁, A ₂ of rank 1 and strongly indecomposable group B of rank 2 such that quasi-homomorphism groups Q ? Hom(A _i , B) and Q ? Hom(B, A _i ) for any i = 1, 2 have rank 1 or are zero. Moreover, for algebras from the classification we present necessary and sufficient conditions for their realization as quasi-endomorphism rings of these groups.     

Homotopy invariants of mappings to the circle

Homotopy classes of mappings of a space X to the circle T form an Abelian group B(X) (the Bruschlinsky group). If a: X → T is a continuous mapping, then [a] denotes the homotopy class of a, and I _r (a): (X × T) _r → \mathbbZ \mathbb{Z} is the indicator function of the rth Cartesian power of the graph of a. Let C be an Abelian group and let f: B(X) → C be a mapping. By definition, f has order not greater than r if the correspondence I _r (a) → f([a]) extends to a (partly defined) homomorphism from the Abelian group of Z-valued functions on (X × T) ^r to C. It is proved that the order of f equals the algebraic degree of f. (A mapping between Abelian groups has degree at most r if all of its finite differences of order r +1 vanish.) Bibliography: 2 titles.     

The group Hom(<Emphasis Type="Italic">A,B</Emphasis>) as an Artinian E(<Emphasis Type="Italic">B</Emphasis>)-or E(<Emphasis Type="Italic">A</Emphasis>)-module

The conditions of Artinianity of the homomorphism group Hom(A, B) as a module over the endomorphism ring of the Abelian group B or A are found. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 81–96, 2007.     

The divisor class groups of some rings of holomorphic functions

Let (X, _x O) be a normal complex analytic space andAX a connected Stein compact set, i.e. a compact subset ofX which has a basis of open neighborhoods which are Stein spaces. We restrict attention to thoseA such thatR=H ⁰ O) is Noetherian. In Section I various exact sequences involving the divisor class group ofR, denotedC(R), are developed. (IfA is a point, one of these sequences is well-known [24], [39].)LetB be a connected compact Stein set on a normal varietyY such thatS=H ^o(B, _Y O) andT=H ^o(A×B, _X×Y O are Noetherian. In II we give a Künneth-type formula which relatesC(R), C(S) andC(T). In III we show certain analytic local rings are unique factorization domains, study the divisor class groups of local rings on the quotient of an analytic space by a finite group, and prove a simple result on the topology of germs of complex analytic sets. We give a function-theoretic proof that complete intersections of dimensions greater than three which have isolated singularities have local rings which are unique factorization domains.The author wishes to thank Columbia University and the Forschungsinstitut für Mathemathik der ETH (Zürich) for their hospitality and the NSF for financial support.     

On Chow and Brauer groups of a product of Mumford curves

Let C₁,···,C_d be Mumford curves defined over a finite extension of and let X=C₁×···×C_d. We shall show the following: (1) The cycle map CH₀(X)/n → H^2d(X, μ_n^⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH₀(X)→Hom(Br(X),) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH₀(X).     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

Spectra of composition operators on the bloch and bergman spaces

Whenϕ is an analytic map of the unit diskU into itself, andX is a Banach space of analytic functions onU, define the composition operatorC _ϕ byC _ϕ (f)=f o ϕ, forf∈X. In this paper we show how to use the Calderón theory of complex interpolation to obtain information on the spectrum ofC _ϕ (under suitable hypotheses onϕ) acting on the Bloch spaceB and BMOA, the space of analytic functions in BMO. To do this we first obtain some results on the essential spectral radius and spectrum ofC _ϕ on the Bergman spacesA ^pand Hardy spacesH ^p,spaces which are connected toB and BMOA by the interpolation relationships [A ¹,B] _t =A ^pand [H ¹,BMOA] _t =H ^pfor 1=p(1−t).     

On ascending and subnormal subgroups of infinite factorized groups

We consider an almost hyper-Abellan group G of a finite Abelian sectional rank that is the product of two subgroups A and B. We prove that every subgroup H that belongs to the intersection A ∩ B and is ascending both in A and B is also an ascending subgroup in the group G. We also show that, in the general case, this statement is not true. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 6, pp. 842–848, June, 1997.     

C-semigroups and strongly continuous semigroups

We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA| _Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖_[Im(C)]≡‖C ⁻¹ x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B| _X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true. We construct fractional powers of generators of boundedC-semigroups. We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.     

A characterization of alternating groups. II

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A₄ or to A₅. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A₄. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A₅ then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈X^G〉 is a direct product of subgroups 〈C _α〉-generated by some connected components C _α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A₅. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A₅ or is cyclic of order 5. Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006.     

Sums of lengtht in Abelian groups

LetG be an Abelian group written additively,B a finite subset ofG, and lett be a positive integer. Fort≦|B|, letB _t denote the set of sums oft distinct elements overB. Furthermore, letK be a subgroup ofG and let σ denote the canonical homomorphism σ:G→G/K. WriteB _t (modB _t) forB _tσ and writeB _t (modK) forBσ. The following addition theorem in groups is proved. LetG be an Abelian group with no 2-torsion and letB a be finite subset ofG. Ift is a positive integer such thatt<|B| then |B _t (modK)|≧|B (modK)| for any finite subgroupK ofG.     

On the Perturbations of Regular Linear Systems and Linear Systems with State and Output Delays

This paper is concerned with perturbation problems of regularity linear systems. Two types of perturbation results are proved: (i) the perturbed system (A + P, B, C) generates a regular linear system provided both (A, B, C) and (A, B, P) generate regular linear systems; and (ii) the perturbed system ((A_-1+DA)|_X,B,C^A_L){((A_{-1}+\Delta A)|_X,B,C^A_\Lambda)} generates a regular linear system if both (A, B, C) and (A, ΔA, C) generate regular linear systems. These allow us to establish a new variation of constants formula of the control system (A + P, B). Moreover, these results are applied to the linear systems with state and output delays. The regularity and the mild expressibility is deduced, and a necessary and sufficient condition for stabilizability of the delayed systems is proved.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

Order of a function on the Bruschlinsky group of a two-dimensional polyhedron

Homotopy classes of mappings of a compact polyhedron X to the circle T form an Abelian group B(X), which is called the Bruschlinsky group and is cananically isomorphic to H ¹ (X; ℤ), Let L be an Abelian group, and let f: B(X) → L be a function. One says that the order of f does not exceed r if for each mapping a: X → T the value f([a]) is ℤ-linearly expressed via the characteristic function I _r (a): (X × T) ^r → ℤ of (Γ _a ) ^r , where Γ _a ⊂ X × T is the graph of a. The (algebraic) degree of f is not greater than r if the finite differences of f of order r + 1 vanish. Conjecturally, the order of f is equal to the algebraic degree of f. The conjecture is proved in the case where dim X ≤ 2. Bibliography: 1 title.     

On pureness in Abelian groups

Torsion-free Abelian groups G and H are called quasi-equal (G ≈ H) if λG ⊂ H ⊂ G for a certain natural number ≈. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups; here a group is considered as a left module over its endomorphism ring. On the other hand, a topical problem in the Abelian group theory is the problem of investigation of pureness in the category of Abelian groups (see [4]). We consider the pureness introduced by P. Cohn [2] for Abelian groups as modules over their endomorphism rings. Particularity of the investigation of the properties of pureness for the Abelian group G as the module _E (G)G lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring R with the identity element. Indeed, if _R M is an arbitrary unitary left module and M ⁺ is its Abelian group, then each element from R can be identified with an appropriate endomorphism from the ring E(M ⁺) under the canonical ring homomorphism R → E(M ⁺). Then it holds that if _E(M+) N is a pure submodule in _E(M+) M ⁺, then _R N is a pure submodule in _R M. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 225–238, 2004.