A note on the tensor product of Lie soluble algebras

D. M. Riley proved in [3] that, if A and B are either Lie nilpotent or Lie metabelian algebras, then their tensor product A ⊗ B is Lie soluble and obtained bounds on the Lie derived length of A ⊗ B. The aim of the present note is to improve Riley’s bounds; moreover we consider also the cases in which A and B are either strongly Lie soluble or strongly Lie nilpotent algebras. Received: 5 April 2006 The first two authors partially supported by MIUR-Italy via PRIN “Group theory and applications”.     

<Emphasis Type="Italic">C</Emphasis>*-Independence,Product States and Commutation

Let D be a unital C*-algebra generated by C*-subalgebras A and B possessing the unit of D. Motivated by the commutation problem of C*-independent algebras arising in quantum field theory, the interplay between commutation phenomena, product type extensions of pairs of states and tensor product structure is studied. Rooss theorem [11] is generalized in showing that the following conditions are equivalent: (i) every pair of states on A and B extends to an uncoupled product state on D; (ii) there is a representation of D such that (A) and (B) commute and is faithful on both A and B; (iii) is canonically isomorphic to a quotient of D.The main results involve unique common extensions of pairs of states. One consequence of a general theorem proved is that, in conjunction with the unique product state extension property, the existence of a faithful family of product states forces commutation. Another is that if D is simple and has the unique product extension property across A and B then the latter C*-algebras must commute and D be their minimal tensor product.Communicated by Klaus Fredenhagensubmitted 03/12/03, accepted 26/04/04     

Minimum recession-compatible subsets of closed convex sets

A subset B of a closed convex set A is recession-compatible with respect to A if A can be expressed as the Minkowski sum of B and the recession cone of A. We show that if A contains no line, then there exists a recession-compatible subset of A that is minimal with respect to set inclusion. The proof only uses basic facts of convex analysis and does not depend on Zorn’s Lemma. An application of this result to the error bound theory in optimization is presented.     

The semilattice tensor product of projective distributive lattices

Grant A. Fraser defined the semilattice tensor productA ⊗B of distributive latticesA, B and showed that it is a distributive lattice. He proved that ifA ⊗B is projective then so areA andB, that ifA andB are finite and projective thenA ⊗B is projective, and he gave two infinite projective distributive lattices whose semilattice tensor product is not projective. We extend these results by proving that ifA andB are distributive lattices with more than one element thenA ⊗B is projective if and only if bothA andB are projective and both have a greatest element. Presented by W. Taylor.     

Continuity of the straight line path for convex sets

IfA andB are closed nonempty sets in a locally convex space, the straight line path fromA toB is defined by the formulaφ(α)=cl (αA+(1−α)B), 0≦α≦1. IfA andB are convex, then continuity of the path with respect to the Hausdorff uniform topology is necessary for both connectedness and path connectedness ofA toB within the convex sets so topologized. We also produce internal necessary and sufficient conditions for continuity of the path between pairs of convex sets.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

On the composition of operators in vector spaces and their Noether property

Under the minimal assumptions imposed on given spaces, the sufficient conditions are established, under which the compositionBA of operatorsA andB has the Noether property and is normally solvable. Similar conditions, guaranteeing that the operatorA is normally solvable or possesses the Noether property, are obtained for the operatorsB andBA.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1177–1180, August, 1993.     

On subgroups of finite <Emphasis Type="Italic">p</Emphasis>-groups

In §2, we prove that if a 2-group G and all its nonabelian maximal sub-groups are two-generator, then G is either metacyclic or minimal non-abelian. In §3, we consider a similar question for p > 2. In §4 the 2-groups all of whose minimal nonabelian subgroups have order 16 and a cyclic subgroup of index 2, are classified. It is proved, in §5, that if G is a nonmetacyclic two-generator 2-group and A, B, C are all its maximal subgroups with d(A) ≤ d(B) ≤ d(C), then d(C) = 3 and either d(A) = d(B) = 3 (this occurs if and only if G/G′ has no cyclic subgroup of index 2) or else d(A) = d(B) = 2. Some information on the last case is obtained in Theorem 5.3.     

An approximate,multivariable version of Specht's theorem

In this article we provide generalizations of Specht's theorem which states that two n × n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A?) and (B, B?) coincide. First, we obtain conditions which allow us to extend this to simultaneous similarity or unitary equivalence of families of operators, and secondly, we show that it suffices to consider a more restricted family of functions when comparing traces. Our results do not require the traces of words in (A, A?) and (B, B?) to coincide, but only to be close.     

Minimal elementary extensions of models of set theory and arithmetic

We discuss minimal elementary extensions of models of set theory and contrast the behavior of models of set theory and arithmetic as regarding such extensions. Our main result, proved using a Boolean ultrapower argument, is:Theorem Every model of ZFChas a conservative elementary extension which possesses a cofinal minimal elementary extension.An application of Boolean ultrapowers to models of full arithmetic is also presented.The results of this paper were presented at the spring meeting of the Association for Symbolic Logic held at Pennsylvania State University during April 7 and 8, 1990     

Adjoining an Identity to a Reduced Archimedean f-Ring

Let frA denote the category of f-rings which are reduced and Archimedean, and let Φ be the (nonfull) subcategory of such rings with identity (each with the natural morphisms). Some time ago, the second author showed, using his representation theory, that for each A ∈ | frA| there is a certain minimal embedding u _A :A→ uA ∈ | Φ|. More recently, he has revisited the representation theory, expanding it to include the representation of morphisms. Based upon this, the present article analyzes the operator u:| frA| → Φ: the construction of uA is tidied, several characterizations of the pair (u _A , uA) are given, and the relation between the maximal ideal structures of A and uA is described. Membership in the class U of frA-morphisms that are “u-extendable” is characterized and it is shown that U = (| frA|,U) is a category in which Φ is a full essentially-reflective subcategory. The frA-objects are characterized for which, respectively, ? B(frA(A, B) = U (A, B)), and, ? B ≠ 0(frA(B, A) = U(B, A)).     

Edge disjoint Steiner trees in graphs without large bridges

A set A of vertices of an undirected graph G is called k‐edge‐connected in G if for all pairs of distinct vertices a, b∈A, there exist k edge disjoint a, b‐paths in G. An A‐tree is a subtree of G containing A, and an A‐bridge is a subgraph B of G which is either formed by a single edge with both end vertices in A or formed by the set of edges incident with the vertices of some component of G ? A. It is proved that (i) if A is k·(? + 2)‐edge‐connected in G and every A‐bridge has at most ? vertices in V(G) ? A or at most ? + 2 vertices in A then there exist k edge disjoint A‐trees, and that (ii) if A is k‐edge‐connected in G and B is an A‐bridge such that B is a tree and every vertex in V(B) ? A has degree 3 then either A is k‐edge‐connected in G ? e for some e∈E(B) or A is (k ? 1)‐edge‐connected in G ? E(B). © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 188–198, 2009     

Rational obstruction theory and rational homotopy sets

We develop an obstruction theory for homotopy of homomorphisms between minimal differential graded algebras. We assume that has an obstruction decomposition given by and that f and g are homotopic on . An obstruction is then obtained as a vector space homomorphism . We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes . This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps. Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000     

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct. Presented by K. Goodearl.     

▵-Reductions of modules

Let △ be a multiplicatively closed set of finitely generated nonzero ideals of a ring R. Then the concept of a △ -reduction of an R -submodule D of an R -module A is introduced and several basic properties of such reductions are established. Among these are that a minimal △ -reduction B of D exists and that every minimal basis of B can be extended to a minimal basis of all R -submodules between B and D, when R is local and A is a finite R -module. Then, as an application, △ -reductions B of a submodule C with property (^?) are introduced, characterized, and shown to be quite plentiful. Here, (^?) means that (R ,M) is a local ring of altitude at least one, that △ = {M_n ; n ≥ 0} and that if D ? E are R -submodules between B and C, then every minimal basis of D can be extended to a minimal basis of E.     

Depth Two,Normality, and a Trace Ideal Condition for Frobenius Extensions

A ring extension A ‖ B is depth two if its tensor-square satisfies a projectivity condition w.r.t. the bimodules _A A _B and _B A _A . In this case the structures (A ? _B A) ^B and End  _B A _B are bialgebroids over the centralizer C _A (B) and there is a certain Galois theory associated to the extension and its endomorphism ring. We specialize the notion of depth two to induced representations of semisimple algebras and character theory of finite groups. We show that depth two subgroups over the complex numbers are normal subgroups. As a converse, we observe that normal Hopf subalgebras over a field are depth two extensions. A generalized Miyashita–Ulbrich action on the centralizer of a ring extension is introduced, and applied to a study of depth two and separable extensions, which yields new characterizations of separable and H-separable extensions. With a view to the problem of when separable extensions are Frobenius, we supply a trace ideal condition for when a ring extension is Frobenius.     

Abelian groups with normal endomorphism rings

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group A with a normal endomorphism ring contains a pure fully invariant subgroup G ⊕ B, the endomorphism ring of a group G is commutative, and a subgroup B is not always distinguished by a direct summand in A. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.     

Some connected ramsey numbers

A graph G is co-connected if both G and its complement ? are connected and nontrivial. For two graphs A and B, the connected Ramsey number r_c(A, B) is the smallest integer n such that there exists a co-connected graph of order n, and if G is a co-connected graph on at least n vertices, then A ? G or B ? ?. If neither A or B contains a bridge, then it is known that r_c(A, B) = r(A, B), where r(A, B) denotes the usual Ramsey number of A and B. In this paper r_c(A, B) is calculated for some pairs (A, B) when r(A, B) is known and at least one of the graphs A or B has a bridge. In particular, r_c(A, B) is calculated for A a path and B either a cycle, star, or complete graph, and for A a star and B a complete graph.     

On the Perturbations of Regular Linear Systems and Linear Systems with State and Output Delays

This paper is concerned with perturbation problems of regularity linear systems. Two types of perturbation results are proved: (i) the perturbed system (A + P, B, C) generates a regular linear system provided both (A, B, C) and (A, B, P) generate regular linear systems; and (ii) the perturbed system ((A_-1+DA)|_X,B,C^A_L){((A_{-1}+\Delta A)|_X,B,C^A_\Lambda)} generates a regular linear system if both (A, B, C) and (A, ΔA, C) generate regular linear systems. These allow us to establish a new variation of constants formula of the control system (A + P, B). Moreover, these results are applied to the linear systems with state and output delays. The regularity and the mild expressibility is deduced, and a necessary and sufficient condition for stabilizability of the delayed systems is proved.     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).