Some new results on functions in C(X) having their support on ideals of closed sets

Abstract

For any ideal of closed sets in X, let be the family of those functions in C(X) whose support lie on . Further let contain precisely those functions f in C(X) for which for each ? > 0, {x ∈ X: |f (x)| ≥ ?} is a member of . Let stand for the set of all those points p in βX at which the stone extension f^? for each f in is real valued. We show that each realcompact space lying between X and βX is of the form if and only if X is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally- or almost locally-, becomes a space of the same form. We further show that is a free ideal (essential ideal) of C(X) if and only if is a free ideal (essential ideal) of when and only when X is locally- (almost locally-). We address the problem, when does or become identical to the socle of the ring C(X). The results obtained turn out to imply a special version of the fact obtained by Azarpanah corresponding to the choice ≡ the ideal of compact sets in X. Finally we observe that the ideals of the form of C(X) are no other than the z^?-ideals of C(X).     

The Schur Lie-multiplier of Leibniz algebras

Abstract

For a free presentation 0 → τ → → → 0 of a Leibniz algebra , the Baer invariant is called the Schur multiplier of relative to the Liezation functor or Schur Lie-multiplier. For a two-sided ideal of a Leibniz algebra , we construct a four-term exact sequence relating the Schur Lie-multipliers of and /, which is applied to study and characterize Lie-nilpotency, Lie-stem covers and Lie-capability of Leibniz algebras.     

Multiplicative ∗-lie triple higher derivations of standard operator algebras

Abstract

Let be a standard operator algebra on an infinite dimensional complex Hilbert space containing identity operator I. In this paper it is shown that if is closed under the adjoint operation, then every multiplicative ?-Lie triple derivation is a linear ?-derivation. Moreover, if there exists an operator S ∈ such that S + S^? = 0 then d(U) = U S ? SU for all U ∈ , that is, d is inner. Furthermore, it is also shown that any multiplicative ?-Lie triple higher derivation D = {δ_n}_n∈? of is automatically a linear inner higher derivation on with d(U)^? = d(U^?).     

Ideal convergent subseries in Banach spaces

Abstract

Assume that is an ideal on ?, and ∑_n x_n is a divergent series in a Banach space X. We study the Baire category, and the measure of the set A() := {t ∈ {0, 1}^?: ∑_n t(n)x_n is -convergent}. In the category case, we assume that has the Baire property and ∑_n x_n is not unconditionally convergent, and we deduce that A() is meager. We also study the smallness of A() in the measure case when the Haar probability measure λ on {0, 1}^? is considered. If is analytic or coanalytic, and ∑_n x_n is -divergent, then λ(A()) = 0 which extends the theorem of Dindo?, ?alát and Toma. Generalizing one of their examples, we show that, for every ideal on ?, with the property of long intervals, there is a divergent series of reals such that λ(A(Fin)) = 0 and λ(A()) = 1.     

A study of the order dimension of a poset using matrices

In every finite poset (X, ≤) we assign the so called order-matrix , where α_ij ∈ {?2, 0, 1, 2}. Using this matrix, we characterize the order dimension of an arbitrary finite poset.     

On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

Let SN_r (r ≥ 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix

the inequality

(p = 2; 3;?…?) is proved, where λ_k(A) (k = 1; 2;?…?) are the eigenvalues of A and N_r(.) is the norm in SN_r. Moreover, let P(z) = z²I + Bz + C (z ∈ ?) with B ∈ SN_2p, C ∈ SN_p. By z_k(P) (k = 1; 2;?…?) the characteristic values of the pencil P are denoted. It is shown that

In the case p = 1, sharper results are established. In addition, it is derived that

     

Two point-picking games derived from a property of function spaces

We present a study of two versions of the point-picking game defined by Berner and Juhasz. Given a space X there are two rivals O and P who take turns playing on X. In the n-th round Player O takes a non-empty open subset U_n of the space X and P responds by choosing a point x_n ∈ U_n. After ω-many moves are completed, the family is called the play of the game. In the CD-game CD(X) Player P wins if the set is closed and discrete. Otherwise O is the winner. In the CL-game CL(X, p), where the point p ∈ X is fixed, Player O wins if contains p in its closure. If , then P is declared to be the winner. We show that in spaces C_p(X) both CD-game and CL-game are equivalent to Gruenhage’s W-game for Player O. If , then Player O has a winning strategy in CL(X, p). The converse is not always true. However, if X is separable or compact of π-weight ≤ ω₁, then existence of a winning strategy for O in CL(X, p) is equivalent to .     

Remarks on ω-domination of discrete subspaces

Abstract

Given a space X, we will say that a class of subsets of X is dominated by a class ? if for any A ∈ , there exists a B ∈ ? such that A ? . In particular, all (closed) discrete subsets of X are countably dominated (which we frequently abbreviate as ω-dominated) if, for any (closed) discrete set D ? X, there exists a countable set B ? X such that D ? . In this paper, we investigate the topological properties of spaces in which (closed) discrete subspaces are dominated either by countable subsets or by Lindelöf subspaces.     

On Amitsur rings

Abstract

In this note, we introduce (hereditary) Amitsur rings and give examples of (hereditary) Amitsur rings. We construct radicals and _S. We also find radicals in which every prime ring is a hereditary Amitsur ring and radicals in which every prime ring is not a hereditary Amitsur ring. We give characterizations for (hereditary) Amitsur rings and prove that the semisimple class S_S is polynomial extensible. We show that all zero rings are Amitsur rings and all Baer radical rings are hereditary Amitsur rings.     

Strict and Mackey topologies and tight vector measures

Let B() denote the Banach algebra of all bounded Borel measurable complex functions defined on a topological Hausdor? space X, and B_o() stand for the ideal of B() consisting of all functions vanishing at infinity. Then B() is a faithful Banach left B_o()-module and the strict topology β on B() induced by B_o() is a mixed topology. For a sequentially complete locally convex Hausdor? space (E, ξ), we study the relationship between vector measures m : → E and the corresponding continuous integration operators T_m : B() → E. It is shown that a measure m : → E is countably additive tight if and only if the corresponding integration operator T_m is (η, ξ)-continuous, where η denotes the infimum of the strict topology β and the Mackey topology τ (B(), ca()). If, in particular, E is a Banach space, it is shown that m is countably additive tight if and only if Tm(absconv(U ∪ W)) is relatively weakly compact in E for some τ (B(), ca())-neighborhood U of 0 and some β-neighborhood W of 0 in B(). As an application, we prove a Nikodym type convergence theorem for countably additive tight vector measures.     

Ordering n-vertex cacti with matching number q by their spectral radii

Abstract

A connected graph G is a cactus if any two of its cycles have at most one common vertex. Denote by the set of n-vertex cacti with matching number q. Huang, Deng and Simi? [23] identified the unique graph with the maximum spectral radius among 2q-vertex cacti with perfect matchings. In this paper, as a continuance of it, the largest and second largest spectral radii together with the corresponding graphs among are determined. Consequently, the first two largest spectral radii together with cacti having perfect matchings are also determined.     

Remarks on star-C-Menger spaces

In this paper, we show that assuming ω₁ < , there exists a Tychono? star-C-Menger space having a regular-closed G_δ-subspace which is not star-C-Menger which gives a partial answer to a question of Song and Yin [12], and continue to investigate topological properties of star-C-Menger spaces.