Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

On δ-I- open sets and decomposition of α-I-continuity

We introduce the notions of δ-I- open sets and semi δ-I-continuous functions in ideal topological spaces and investigate some of their properties. Additionally, we obtain decompositions of semi-I-continuous functions and α-I-continuous functions by using δ-I-open sets. This revised version was published online in June 2006 with corrections to the Cover Date.     

A decomposition of α-continuity and semi-continuity

The following two decomposition theorems are obtained. (1) A function f is α-continuous if and only if f is pre-continuous and αα-continuous, (2) A function f is semi-continuous if and only if f is spr-continuous and αLC-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

A criterion for the primeness of ideals generated by polynomials with separated variables

Proving primeness of an idealI=〈f ₁, …,f _m〉 in a polynomial ringR=K[X ₁, …,X _n]ofn indeterminates over an algebraically closed fieldK is a difficult task in general. Although there are straightforward algorithms that decide whetherI is prime or not, they are prohibitively lengthy if the number of indeterminates or the degrees of thef _iare large. In this paper we will give an easy criterion for the primeness ofI if thef _iare polynomials with separated variables, i.e. no mixed monomials occur in thef _i. The work on this paper was done while the author was a MINERVA fellow at Tel Aviv University.     

Relative version of the Titchmarsh convolution theorem

We consider the algebra C _u = C _u (ℝ) of uniformly continuous bounded complex functions on the real line ℝ with pointwise operations and sup-norm. Let I be a closed ideal in C _u invariant with respect to translations, and let ah _I (f) denote the minimal real number (if it exists) satisfying the following condition. If λ > ah _I (f), then ( [^(f)] - [^(g)] ) |_V = 0\left. {\left( {\hat f - \hat g} \right)} \right|_V = 0 for some g ∈ I, where V is a neighborhood of the point λ. The classical Titchmarsh convolution theorem is equivalent to the equality ah _I (f ₁ · f ₂) = ah _I (f ₁) + ah _I (f ₂), where I = {0}. We show that, for ideals I of general form, this equality does not generally hold, but ah _I (f ⁿ ) = n · ah _I (f) holds for any I. We present many nontrivial ideals for which the general form of the Titchmarsh theorem is true.     

Generalized derivations with Engel condition on multilinear polynomials

Let R be a prime ring with extended centroid C, δ a nonzero generalized derivation of R, f(x ₁, ..., x _n ) a nonzero multilinear polynomial over C, I a nonzero right ideal of R and k ≥ a fixed integer. If [δ(f(r ₁, ..., r _n )), f(r ₁, ..., r _n )] _k = 0, for all r ₁, ..., r _n ∈ I, then either δ(x) = ax, with (a-γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds (1) if char(R) = 0 then f(x ₁, ..., x _n ) is central valued in eRCe (2) if char(R) = p > 0 then is central valued in eRCe, for a suitable s ≥ 0, unless when char(R) = 2 and eRCe satisfies the standard identity s ₄ (3) δ(x) = ax−xb, where (a+b+α)e = 0, for α ∈ C, and f(x ₁, ..., x _n )² is central valued in eRCe.     

Extending modules relative to a torsion theory

An R-module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ-extending module, where τ is a hereditary torsion theory on Mod-R. An R-module M is called type 2 τ-extending if every type 2 τ-closed submodule of M is a direct summand of M. If τ _I is the torsion theory on Mod-R corresponding to an idempotent ideal I of R and M is a type 2 τ _I -extending R-module, then the question of whether or not M/MI is an extending R/I-module is investigated. In particular, for the Goldie torsion theory τ _G we give an example of a module that is type 2 τ _G -extending but not extending.     

A Convenient Subcategory of Tych

A map f:XY between Hausdorff topological spaces is k-continuous if its restriction f| _K to every compact subspace K of X is continuous. X is called a k _R -space if every k-continuous function from X to a Tychonoff space is continuous. In this paper we investigate the category of Tychonoff k _R -spaces, and show that it is Cartesian closed (thus convenient in the sense of Wyler).     

A note on <Emphasis Type="Italic">f</Emphasis><Subscript><Emphasis Type="Italic">I</Emphasis></Subscript>-sets and <Emphasis Type="Italic">f</Emphasis><Subscript><Emphasis Type="Italic">I</Emphasis></Subscript> -continuous functions

Summary In [6] we introduced and investigated the notions of f_I -sets and f_I -continuous functions in ideal topological spaces. In this paper, we investigate their further important properties.     

On decomposition of A-continuity

Summary The following two theorems are obtained. (1) A mapping fis A-continuous if and only if it is semi-continuous and C-continuous. (2) A mapping f is A-continuous if and only if it is spr-continuous and LC-continuous.     

Mappings preserving some geometrical figures

We introduce new characterizations of linear isometries. More precisely, we prove that if a one-to-one mapping f : Rⁿ →Rⁿ (n > 1) maps the periphery of every regular triangle (quadrilateral or hexagon) of side length a > 0 onto the periphery of a figure of the same type with side length b > 0, then there exists a linear isometry I : Rⁿ →Rⁿ up to translation such that f(x) = (b/a) I(x). This revised version was published online in June 2006 with corrections to the Cover Date.     

Partial cancellation of modules

We investigate partial cancellation of modules and show that if an ideal I of an exchange ring R has stable range one, then A⊕B≌A⊕C implies B≌C for all A∈FP (I). The converse is true when R is a regular ring. For an ideal I of a regular ring, we also show that I has stable range one if and only if perspectivity is transitive in L(A) for all A∈ FP (I). These give nontrivial generalizations for unit-regularity. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point

For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H ₀ : f≡ 0, (the signal f is zero) against the nonparametric alternative H ₁ : f∈Λ_ɛ where Λ_ɛ is a set of functions on R ¹ of the form Λ_ɛ = {f : f∈?, ϕ(f) ≥ Cψ_ɛ}. Here ? is a H?lder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψ_ɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C ^* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C ^* and is not possible for C < C ^*. We propose asymptotically minimax test statistics. Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

On a graph of ideals

We associate a graph Γ₊(R) to a ring R whose vertices are nonzero proper right ideals of R and two vertices I and J are adjacent if I+J=R. Then we try to translate properties of this graph into algebraic properties of R and vice versa. For example, we characterize rings R for which Γ₊(R) respectively is connected, complete, planar, complemented or a forest. Also we find the dominating number of Γ₊(R).     

Punishing Factors for Finitely Connected Domains

Let Ω and Π be two finitely connected hyperbolic domains in the complex plane \Bbb C{\Bbb C} and let R(z, Ω) denote the hyperbolic radius of Ω at z and R(w, Π) the hyperbolic radius of Π at w. We consider functions f that are analytic in Ω and such that all values f(z) lie in the domain Π. This set of analytic functions is denoted by A(Ω, Π). We prove among other things that the quantities C_n(W,P) := sup_{f ? A(W,P)}sup_{z ? W}\frac|f⁽ⁿ⁾(z)| R(f(z),P)n! (R(z,W))ⁿC_n(\Omega,\Pi)\,:=\,\sup_{f\in A(\Omega,\Pi)}\sup_{z\in \Omega}\frac{\vert f^{(n)}(z)\vert\,R(f(z),\Pi)}{n!\,(R(z,\Omega))^n} are finite for all n ? \Bbb N{n \in {\Bbb N}} if and only if ∂Ω and ∂Π do not contain isolated points.