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 .     

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.     

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.     

Remarks on intermediate C-rings of C(X)

Let X be a Tychono? space and A(X) be a subring of C(X) containing C^?(X). We introduce the notion of -ideal in A(X). It is observed that the class of -ideals contains the class of z_A-ideals and is contained in the class of z-ideals of A(X). These containments may be proper. It turns out that coincidence of z-ideals of A(X) with -ideals characterizes intermediate C-rings of C(X).     

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).     

On countably uniform closed-spaces

Abstract

Let X be a topological space and C_c(X) be the functionally countable subalgbera of C(X). We call X to be a countably uniform closed-space, briefly, a CU C-space, if C_c(X) is closed under uniform convergence. We investigate that countably uniform closedness need not closed under finite intersection and infinite product. It is shown that if X is a countable union of quasi-components, then X is a CU C-space. We characterize C_c-embedding and also -embedding in CU C-spaces. A subset S of X is called Z_c-embedded, if each Z ∈ Z_c(S) is the restriction of a zero-set of Z_c(X). It is observed that in a zero-dimensional CU C-space, each Lindelöf subspae is Z_c-embedded. Moreover, it is shown that in CU C-spaces, each Lindelöf subspace is C_c-embedded if and only if it is c-completely separated from each zero-set, which is disjoint from it. Also in latter spaces, it is observed that for each S ? X, C_c-embedding, -embedding and Z_c-embedding coincide, when S belongs to Z_c(X) or it is a c-pseudocompact space. Finally, when X is both a CU C-space and a CP-space, then each Z_c-embedded subspace is C_c-embedded (-embedded) in X.     

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^?).     

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.     

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.     

On some parallelism between complete regularity and zero-dimensionality

We explore some parallelism between the categories CRFrm and 0DFrm of completely regular frames and zero-dimensional frames, respectively, with a view to establishing zero-dimensional analogues of C*-quotients. A lattice homomorphism between the cozero parts of two completely regular frames can be lifted to a frame homomorphism between the Stone-?ech compactifications of the frames involved [13]. Here we lift a lattice homomorphism ψ: BL → BM between the Boolean parts of two zero-dimensional frames to a frame homomorphism between their universal zero-dimensional compactifications, and then study some properties of the lift.     

Practical numbers in Lucas sequences

Abstract

A practical number is a positive integer n such that all the positive integers m ≤ n can be written as a sum of distinct divisors of n. Let (u_n)_n≥0 be the Lucas sequence satisfying u₀ = 0, u₁ = 1, and u_n+2 = au_n+1 + bu_n for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a² + 4b > 0. Also, let be the set of all positive integers n such that |u_n| is a practical number. Melfi proved that is infinite. We improve this result by showing that #(x) ? x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding .     

Characterizations of generalized derivations associated with hochschild 2-cocycles and higher derivations

Let be a unital Banach algebra and be a unital -bimodule. A bilinear mapping α : is called a Hochschild 2-cocycle if xα(y, z) ? α(xy, z) + α(x, yz) ? α(x, y)z = 0 for any . We show that if δ is a linear mapping from into satisfying δ(xy) = δ(x)y + xδ(y) + α(x, y) for any with xy = W, where is a left or right separating point of , then δ is a generalized Jordan derivation associated with a Hochschild 2-cocycle α. We also find the relation of higher derivations and generalized derivations associated with Hochschild 2-cocycles.     

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