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.     

Contractivity of Projections Commuting with Inner Derivations on JBW*-triples

It is shown that if P is a weak*-continuous projection on a JBW*-triple A with predual A _*, such that the range PA of P is an atomic subtriple with finite-dimensional Cartan-factors, and P is the sum of coordinate projections with respect to a standard grid of PA, then P is contractive if and only if it commutes with all inner derivations of PA. This provides characterizations of 1-complemented elements in a large class of subspaces of A _* in terms of commutation relations.     

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.     

f I-sets and decomposition of R I C-continuity

First, we introduce the notion of f _I-sets and investigate their properties in ideal topological spaces. Then, we also introduce the notions of R _I C-continuous, f _I-continuous and contra*-continuous functions and we show that a function f: (X,τ,I) to (Y,φ) is R _I C -continuous if and only if it is f _I-continuous and contra*-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on localizations of mapping spaces

We show that if A is a simply connected, finite, pointed CW-complex, then the mapping spaces Map_*(A,X) are preserved by the localization functors only if A has the rational homotopy type of a wedge of spheres V _l S ^k .     

On N0-quasi-continuous exchange rings

An associative ring R with identity is said to have stable range one if for any a,b? R with aR + bR = R, there exists y ? R such that a + by is left (equivalently, right) invertible. The main results of this note are Theorem 2: A left or right continuous ring R has stable range one if and only if R is directly finite (i.e xy = 1 implies yx = 1 for all x,y ? R), Theorem 6: A left or right N ₀o-quasi-continuous exchange ring has stable range one if and only if it is directly finite, and Theorem 12: left or right N ₀-quasi-continuous strongly π-regular rings have stable range one. Theorem 6 generalizes a well-known result of Goodearl [10], which says that a directly finite, right N _o-continuous von Neumann regular ring is unit-regular     

F-Continuous Graphs

For a nontrivial connected graph F, the F-degree of a vertex in a graph G is the number of copies of F in G containing . A graph G is F-continuous (or F-degree continuous) if the F-degrees of every two adjacent vertices of G differ by at most 1. All P₃-continuous graphs are determined. It is observed that if G is a nontrivial connected graph that is F-continuous for all nontrivial connected graphs F, then either G is regular or G is a path. In the case of a 2-connected graph F, however, there always exists a regular graph that is not F-continuous. It is also shown that for every graph H and every 2-connected graph F, there exists an F-continuous graph G containing H as an induced subgraph.     

Results on <Emphasis Type="Italic">F</Emphasis>-continuous graphs

For any nontrivial connected graph F and any graph G, the F-degree of a vertex v in G is the number of copies of F in G containing v. G is called F-continuous if and only if the F-degrees of any two adjacent vertices in G differ by at most 1; G is F-regular if the F-degrees of all vertices in G are the same. This paper classifies all P ₄-continuous graphs with girth greater than 3. We show that for any nontrivial connected graph F other than the star K _1,k , k ⩾ 1, there exists a regular graph that is not F-continuous. If F is 2-connected, then there exists a regular F-continuous graph that is not F-regular.      

The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸?, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸?.     

S-continuous functions and certain weak forms of regularity and complete regularity

A topological space is called s-regular if each closed connected set and a point outside it are separated by disjoint open sets. Similarly notion of complete s-regularity is introduced; basic properties of s-regular spaces and completely s-regular spaces are studied and interrelations between them and the standard separation axioms are observed. It is shown that in the class of semilocally connected spaces s-regularity coincides with regularity and complete s-regularity coincides with complete regularity. Moreover, properties of s-continuous functions are studied and it is shown that s-regularity and completely s-regularity are preserved under certain s-continuous mappings.     

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).     

ω‐categorical weakly o‐minimal expansions of Boolean lattices

We study ω‐categorical weakly o‐minimal expansions of Boolean lattices. We show that a structure ?? = (A,≤, ?) expanding a Boolean lattice (A,≤) by a finite sequence I of ideals of A closed under the usual Heyting algebra operations is weakly o‐minimal if and only if it is ω‐categorical, and hence if and only if A/I has only finitely many atoms for every I ∈ ?. We propose other related examples of weakly o‐minimal ω‐categorical models in this framework, and we examine the internal structure of these models.     

Asymptotically spirallike mappings in several complex variables

In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space ℂ ⁿ . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric representation, where A ∈ L(ℂ ⁿ , ℂ ⁿ ), and prove that if dt < ∞ (this integral is convergent if k ₊(A) < 2m(A)), then a mapping f ∈ S(B ⁿ ) is A-asymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f(z, t) such that D f(0, t) = e ^At , {e ^−At f(·, t)} _t≥0 is a normal family on B ⁿ and f = f(·, 0). In particular, a spirallike mapping with respect to A ∈ L(ℂ ⁿ , ℂ ⁿ ) with dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A* = 2I _n , then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings. Partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221. Partially supported by Grant-in-Aid for Scientific Research (C) no. 19540205 from Japan Society for the Promotion of Science, 2007. Partially supported by Romanian Ministry of Education and Research, CEEX Program, Project 2-CEx06-11-10/2006.     

Gantmacher Type Theorems for Holomorphic Mappings

Given a holomorphic mapping of bounded type g ∈ H_b(U, F), where U ? E is a balanced open subset, and E, F are complex Banach spaces, let A : H_b(F) ∈ H_b(U) be the homomorphism defined by A(f) = fog for all f ∈ H_b(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact; (b) A is completely continuous if and only if g(W) is a Dunford-Pettis set for every U-bounded subset W ? U. To obtain these results, we prove that the class of Dunford - Pettis sets is stable under projecti ve tensor products. Moreover, we diaracterize the reflexivity of the space H_b(U,F) and prove that E' and F have the Schur property if and only if H_b(U, F) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings.     

Straight Rings

A (commutative integral) domain is called a straight domain if A ? B is a prime morphism for each overring B of A; a (commutative unital) ring A is called a straight ring if A/P is a straight domain for all P ∈ Spec(A). A domain is a straight ring if and only if it is a straight domain. The class of straight rings sits properly between the class of locally divided rings and the class of going-down rings. An example is given of a two-dimensional going-down domain that is not a straight domain. The classes of straight rings, of locally divided rings, and of going-down rings coincide within the universe of seminormal weak Baer rings (for instance, seminormal domains). The class of straight rings is stable under formation of homomorphic images, rings of fractions, and direct limits. The “straight domain" property passes between domains having the same prime spectrum. Straight domains are characterized within the universe of conducive domains. If A is a domain with a nonzero ideal I and quotient field K, characterizations are given for A ? (I: _K I) to be a prime morphism. If A is a domain and P ∈ Spec(A) such that A _P is a valuation domain, then the CPI-extension C(P) := A + PA _P is a straight domain if and only if A/P is a straight domain. If A is a going-down domain and P ∈ Spec(A), characterizations are given for A ? C(P) to be a prime morphism. Consequences include divided domain-like behavior of arbitrary straight domains.     

TAME TWO-POINT ALGEBRAS WITHOUT LOOPS

We determine the representation type of the algebras whose quiver has precisely two vertices and admits no loops by listing all minimal wild algebras of this form. It turns out that such an algebra A is tame if and only if A/rad³ A is tame, and in this case A degenerates to a special biserial algebra. Moreover, A is wild if and only if it is controlled wild.     

Isometries of JB-algebras

A bijective linear mapping between two JB-algebrasA andB is an isometry if and only if it commutes with the Jordan triple products ofA andB. Other algebraic characterizations of isometries between JB-algebras are given. Derivations on a JB-algebraA are those bounded linear operators onA with zero numerical range. For JB-algebras of selfadjoint operators we have: IfH andK are left Hilbert spaces of dimension ≥3 over the same fieldF (the real, complex, or quaternion numbers), then every surjective real linear isometryf fromS(H) ontoS(K) is of the formf(x)=UoxoU ⁻¹ forx inS(H), whereτ is a real-linear automorphism ofF andU is a real linear isometry fromH ontoK withU(λh)=τ(λ)U(h) for λ inF andh inH. Supported by Acción Integrada Hispano-Alemana HA 94 066 B     

π-Complementation in the Unitisation and Multiplication Algebras of a Semiprime Algebra

π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. For a given semiprime algebra A, we discuss the π-complementation of the unitisation algebra A ¹ of A. Moreover, if in addition the multiplication algebra ?(A) of A is also semiprime, we study the π-complementation in the algebras ?(A) and ?^?(A) (the multiplication ideal of A). In associative setting, we prove that A is π-complemented if and only if ?^?(A) is π-complemented, and that A ¹ π-complemented if and only if ?(A) is π-complemented.     

Quasi-duality for the rings of generalized power series *

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered commutative monoid which is also artinian. For any bimodule _AM_B , we construct a bimodule _A[[S]]M[S]_B[[S]] and prove that _AM_B defines a quasi-duality if and only if the bimodule _A[[S]]M[S]_B[[S]] defines a quasi-duality. As a corollary, it is shown that if a ring A has a quasi-duality then the ring A[[S]] of generalized power series over A has a quasi-duality.