On definability of completely decomposable torsion-free Abelian groups by certain groups of homomorphisms

Let C be an Abelian group. An Abelian group A from a class X of Abelian groups is said to be _C H-definable in X if, for any group B ∈ X, the isomorphism Hom(C,A) ≅ Hom(C,B) implies that A ≅ B. If every group from X is _C H-definable in X, then X is called an _C H-class. In this paper, we study conditions under which a class of completely decomposable torsion-free Abelian groups is an _C H-class, where C is a vector group.     

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.     

When Is the Group Hom(A,B) an Injective E(B)-Module?

Injectivity conditions for the homomorΦsm group Hom(A,B) regarded as a left module over the endomorΦsm ring of the group B are found for arbitrary Abelian groups A and B, where B is nonreduced.     

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.     

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.     

Local selections and local dendrites

Let X be a Peano continuum, C(X) its space of subcontinua, and C(X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C(X) is a continuous choice function; the selection σ is rigid if σ(A) ? B ? A implies σ(A) = σ(B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C(X, ε) admits a selection (rigid selection). Further, C(X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C(X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant.     

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C(X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.     

On derivable mappings

A linear mapping δ from an algebra A into an A-bimodule M is called derivable at c∈A if δ(a)b+aδ(b)=δ(c) for all a,b∈A with ab=c. For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C∈A has a right inverse in B(X) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B(X) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J-subspace lattice algebras, and norm-closed unital standard algebras of B(X). As an application, every Jordan derivation from such an algebra into B(X) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B(X) can be characterized by boundedness and derivability at any fixed C∈A, provided C has a right inverse in B(X). We also show that if A is a canonical subalgebra of an AF C^∗-algebra B and M is a unital Banach A-bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation.     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

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.     

Blockers in hyperspaces

Given a metric continuum X, let X² and C(X) denote the hyperspaces of all nonempty closed subsets and subcontinua, respectively. For A,B∈X² we say that B does not block A if A∩B=∅ and the union of all subcontinua of X intersecting A and contained in X−B is dense in X. In this paper we study some sets of blockers for several kinds of continua. In particular, we determine their Borel classes and, for a large class of locally connected continua X, we recognize them as cap-sets.     

The decomposition of Cr(K) into the direct sum of subalgebras

Let A and B be closed subalgebras of C_r(X) whose direct sum is C_r(X). Some consequences of this relation are explored in this paper. For example if 1 ?A (as may be assumed) it is shown that the norm of the projection onto A is an odd integer and there is a retraction of X onto the set of common zeros of elements of B.     

Controllability,observability and the solution of AX - XB = C

A closed-form finite series representation of the unique solution X of the matrix equation AX ? XB=C is developed. Using this representation, the image, kernel, and rank of X are related to the controllable and unobservable subspaces of the (A, C) and (C, B) pairs respectively. Bounds on the rank of X are obtainedin terms of the dimensions of these subspaces. In the case that C has unitary rank, an exact calculation of rank X is made. The generic rank of X with A, B fixed and C generic is evaluated.     

Solution of fuzzy equations with extended operations

Assuming that 1 is any operation defined on a product set X × Y and taking values on a set Z, it can be extended to fuzzy sets by means of Zadeh's extension principle. Given a fuzzy subset C of Z, it is here shown how to solve the equation $\text{A 1 B = C}$ (or $\text{A 1 B ? C}$) when a fuzzy subset A of X (or a fuzzy subset B of Y) is given. The methodology we provide includes, as a special case, the resolution of fuzzy arithmetical operations, i.e. when 1 stands for +, ?, × or ÷, extended to fuzzy numbers (fuzzy subsets of the real line). The paper is illustrated with several examples in fuzzy arithmetic.     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Sequences of Endomorphism Groups of Abelian Groups

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite p-ranks. For an Abelian group A, a sequence of groups (A_n) is considered, where A₀ = A and A_n+1 = End A_n. It is shown that, if all p-ranks of the group A are finite, then this sequence can stabilize either after A₀ or after A₁.