Topologies on central extensions of von Neumann algebras

Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t _c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t _c(M) restricted to the set E(M) _h of self-adjoint elements of E(M) coincides with the order topology on E(M) _h if and only if M is a σ-finite type I_fin von Neumann algebra.     

On a Bornological Structure in Infinite-Dimensional Holomorphy

The bornology (b) of bounded subsets with respect to continuous convergence is used on spaces of holomorphic functions. It is shown that Hom_coH_b(U) ? U for a circled convex open subset U of a complete nuclear space. Exponential laws for spaces of holomorphic functions with bornological structures are proved and the connection with Colombeau's Silva holomorphic functions is established.     

Commutative lmc algebras with discrete spectrum

SupposeE is a topological algebra with non-empty spectrum Gel'fand spaceM(E) andE M(E)) the algebra of allC-valued continuous functions onM(E). EndowE M(E) with the topologies “c”, “e” of compact resp. equicontinuous convergence. ThenE M (E) characterizes all unital (commutative) semisimple complete lmc algebras with discrete spectrum, while all unital uniform complete lmc algebras with dispersed spectrum are of the formE(E)). The first result may fail if completeness is dropped. The second one fails if “e” is replaced by “c” even ifE is complete. The part of the work due to the 2nd author was carried out during her 3-month visit at the Institute of Mathematics, Univ. of Münster (Germany) in Spring of 1995. The warm hospitality from Professor Dr. G. Maltese and the financial support from a DAAD grant are thankfully acknowledged.     

On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Let H¹(U) denote the space of all pointwise limits of bounded sequences from H(U), where H(U) consists of all continuous functions on the closure [`(U)]\overline{U} of a bounded open set U⊂ℝ^m that are harmonic on U. It is shown that the space H¹(U) is a lattice in the natural ordering if and only if the set ∂_regU of all regular points of U is an F_σ-set.     

A synopsis of Combinatorial Integral Geometry

A Lie coalgebra is a coalgebra whose comultiplication Δ : M → M ? M satisfies the Lie conditions. Just as any algebra A whose multiplication ? : A ? A → A is associative gives rise to an associated Lie algebra $\text{L}$(A), so any coalgebra C whose comultiplication Δ : C → C ? C is associative gives rise to an associated Lie coalgebra $\text{L}$^c(C). The assignment C ? $\text{L}$^c(C) is functorial. A universal coenveloping coalgebra Uc(M) is defined for any Lie Lie coalgebra M by asking for a right adjoint U^c to $\text{L}$^c. This is analogous to defining a universal enveloping algebra U(L) for any Lie algebra L by asking for a left adjoint U to the functor $\text{L}$. In the case of Lie algebras, the unit (i.e., front adjunction) 1 → $\text{L}$ o U of the adjoint functor pair U ? $\text{L}$ is always injective. This follows from the Poincaré-Birkhoff-Witt theorem, and is equivalent to it in characteristic zero (x = 0). It is, therefore, natural to inquire about the counit (i.e., back adjunction) $\text{L}$^c o U^c → 1 of the adjoint functor pair $\text{L}$^c ? U^c.Theorem. For any Lie coalgebra M, the natural map$\text{L}$^c(U^cM) → M is surjective if and only if M is locally finite, (i.e., each element of M lies in a finite dimensional sub Lie coalgebra of M).An example is given of a non locally finite Lie coalgebra. The existence of such an example is surprising since any coalgebra C whose diagonal Δ is associative is necessarily locally finite by a result of that theory. The present paper concludes with a development of an analog of the Poincaré-Birkhoff-Witt theorem for Lie algebras which we choose to call the Dual Poincaré-Birkhoff-Witt Theorem and abbreviate by “The Dual PBWθ.” The constraints of the present paper, however, allow only a sketch of this theorem. A complete proof will appear in a subsequent paper. The reader may, however, consult [12], in the meantime, for details. The Dual PBWθ shows for any locally finite Lie coalgebra M the existence (in χ = 0) of a natural isomorphism of the graded Hopf algebras ₀E(U^cM) and ₀E(S^cM) associated to U^cM and to S^cM = U^c(TrivM) when U^c(M) and S^c(M) are given the Lie filtrations. [Just as U^c(M) is the analog of the enveloping algebra U(L) of a Lie algebra L, so S^c(V) is the analog of the symmetric algebra S(V) on a vector space V. Triv(M) denotes the trivial Lie coalgebra structure on the underlying vector space of M obtained by taking the comultiplication to be the zero map.]     

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.     

Secondary representation of some modules over a commutative ring

Let R be a commutative ring and let M be an R-module with the property that its zero submodule has a primary decomposition. Let E be an injective R-module with W.Ass _R (E) = Ass _R (E) (here W.Ass _R (E) denotes the set of weakly associated primes of E). Then we will show that Hom _R (M,E) has a secondary representation and we will specify the set of its attached prime ideals. This revised version was published online in June 2006 with corrections to the Cover Date.     

Calculus of Functorial Morphisms

We prove that, if F, G: 𝒞 → 𝒟 are two right exact functors between two Grothendieck categories such that they commute with coproducts and U is a generator of 𝒞, then there is a bijection between Nat(F, G) and the centralizer of Hom_𝒟(F(U), G(U)) considered as an Hom_𝒞(U, U)-Hom_𝒞(U, U)-bimodule. We also prove a dual of this result and give applications to Frobenius functors between Grothendieck categories.     

Gorenstein injective modules and Auslander categories

In this paper, we study Gorenstein injective modules over a local Noetherian ring R. For an R-module M, we show that M is Gorenstein injective if and only if Hom _R (Ȓ,M) belongs to Auslander category B(Ȓ), M is cotorsion and Ext ⁱ _R (E,M) = 0 for all injective R-modules E and all i > 0. Received: 24 August 2006 Revised: 30 October 2006     

Regularity and Substructures of Hom

ABSTRACT

We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom _R (A,M) is regular if and only if Ker(f) ?^⊕ A and Im(f) ?^⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom _R (A,M). We study regularity in Hom _R (A ₁ ⊕ A ₂, M ₁ ⊕ M ₂). The existence of a regular function Hom _R (A,M) implies the existence of projective summands of Hom _R (A,M) _{End R (A)} and of _{End R ( M )} Hom _R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.     

Chain Lengths in Certain Random Directed Graphs

We study the random directed graph with vertex set {1, …, n} in which the directed edges (i, j) occur independently with probability c_n/n for i<j and probability zero for i ? j. Let M_n (resp., L_n) denote the length of the longest path (resp., longest path starting from vertex 1). When c_n is bounded away from 0 and ∞ as n→∞, the asymptotic behavior of M_n was analyzed in previous work of the author and J. E. Cohen. Here, all restrictions on c_n are eliminated and the asymptotic behavior of L_n is also obtained. In particular, if c_n/ln(n)→∞ while c_n/n→0, then both M_n/c_n and L_n/c_n are shown to converge in probability to the constant e.     

Recognition of finite groups by a set of orders of their elements

For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups PGL_n(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S₄(7) is recognizable, whereas A₁₀, U₃(5), U₃(7), U₄(2), and U₅(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of its order is at most 11. Then one of the following holds: (i) G is isomorphic to A₅, A₇, A₈, A₉, A₁₁, A₁₂, L₂(q), q=7, 8, 11, 49, L₃(4), S₄(7), U₄(3), U₆(2), M₁₁, M₁₂, M₂₂, HS, or M^cL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A₆, A₁₀, U₃(3), U₄(2), U₅(2), U₃(5), or J₂, and G is nonrecognizable; (iii) G is isomorphic to S₆(2) or O ₈ ⁺ (2), and h(ω(G))=2. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998.     

The unitary group over the integers of an unramified local field

Let L be a lattice over the integers of a local field F which has a nontrivial involution. Then U⁺(L) (the subgroup of rotations of the unitary group U(L)) is generated by unitary transvections and quasitransvections contained in U⁺(L) (Theorem 7.8). Let g be a tableau. Then the mixed commutator subgroup of U⁺(L) and U(g) (the congruence subgroup of U⁺(L) corresponding to g) equals E(g) (the subgroup generated by unitary transvections and quasitransvections with orders contained in g) (Theorem 7.7). Finally, let G be a subgroup of U⁺(L) with _o(G) = g, then G is a normal subgroup of U⁺(L) if and only if U(g) G E(g).     

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, L ? Pic(X) L \in \textrm{Pic}(X) and H an ample divisor on X. Set M_X,H(2; L, c₂) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c₂(F) = c₂. In this paper, we show that the geometry of X and of M_X,H(2; L, c₂) are closely related. More precisely, we prove that for any ample divisor H on X and any L ? Pic(X) L \in \textrm{Pic}(X) , there exists n₀ ? \mathbbZ n_0 \in \mathbb{Z} such that for all n₀ \leqq c₂ ? \mathbbZ n_0 \leqq c_2 \in \mathbb{Z} , M_X,H(2; L, c₂) is rational if and only if X is rational.     

Solution to König's Graph Embedding Problem

Let H : L_p ( R ) → L_p( R ), 1 < p < ∞ be the real HILBERT transform. A bounded, linear operator u:E → F (E, F BANACH spaces) is a HT-operator, if the mapping u ? H : E ? L₂( R , E) → L₂( R , F) has a bounded, linear extension to L₂( R ) → L₂( R , F). For E = F and u = id_E BOURGAIN [3] and BURKHOLDER [5] have shown that this holds if and only if E ? UMD. We study these HT-operators and, in particular, we construct a HT-operator which is not UMD-factorable. Furthermore, we show that a UMD-space E is a HILBERT space if and only if |id_E ? H| = 1.     

Pseudo-finite dimensional representations of $ sl(2,k) $

Let k be an algebraically closed field of characteristic zero and L = sl(2,k) the Lie algebra of 2 × 2 traceless matrices over k. It is shown that there exists a von Neumann regular extension U(L) í U￠(L) U(L) \subseteq U'(L) of the universal enveloping algebra, which is an epimorphism in the category of rings. The article is devoted to the study of the simple representations of U'(L), which may be topologized via the Ziegler topology on the set of injective indecomposable representations of U'(L) or via the Jacobson topology on the set of primitive ideals. These two topologies coincide and the finite dimensional simple representations of L form a dense, discrete and open subset. The field of fractions K(L) of the universal enveloping algebra is another simple representation of U'(L). If the point K(L) is removed from the Ziegler spectrum of U'(L), one obtains a compact totally disconnected topological space, which has the cardinality of the continuum. It is also shown that the lattice of ideals of U'(L) is isomorphic to the lattice of open subsets. The epimorphic ring extension U(L) í U￠(L) U(L) \subseteq U'(L) is used to find an axiomatization of the finite dimensional representations of L in the language of left U(L)-modules. A representation V of L is called pseudo-finite dimensional if it satisfies these axioms. It is shown that a representation V of L is pseudo-finite dimensional if and only if for every central idempotent e ? U￠(L) e \in U'(L) for which eK(L) 1 0 eK(L) \neq 0 , whenever the subrepresentation eV is nonzero, then it has a nonzero highest weight space.     

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.     

On the random 2‐stage minimum spanning tree

It is known [A. M. Frieze, Discrete Appl Math 10 (1985), 47–56] that if the edge costs of the complete graph K_n are independent random variables, uniformly distributed between 0 and 1, then the expected cost of the minimum spanning tree is asymptotically equal to . Here we consider the following stochastic two‐stage version of this optimization problem. There are two sets of edge costs c_M: E → ? and c_T: E → ?, called Monday's prices and Tuesday's prices, respectively. For each edge e, both costs c_M(e) and c_T(e) are independent random variables, uniformly distributed in [0, 1]. The Monday costs are revealed first. The algorithm has to decide on Monday for each edge e whether to buy it at Monday's price c_M(e), or to wait until its Tuesday price c_T(e) appears. The set of edges X_M bought on Monday is then completed by the set of edges X_T bought on Tuesday to form a spanning tree. If both Monday's and Tuesday's prices were revealed simultaneously, then the optimal solution would have expected cost ζ(3)/2 + o(1). We show that, in the case of two‐stage optimization, the expected value of the optimal cost exceeds ζ(3)/2 by an absolute constant ε > 0. We also consider a threshold heuristic, where the algorithm buys on Monday only edges of cost less than α and completes them on Tuesday in an optimal way, and show that the optimal choice for α is α = 1/n with the expected cost ζ(3) ? 1/2 + o(1). The threshold heuristic is shown to be sub‐optimal. Finally we discuss the directed version of the problem, where the task is to construct a spanning out‐arborescence rooted at a fixed vertex r, and show, somewhat surprisingly, that in this case a simple variant of the threshold heuristic gives the asymptotically optimal value 1 ? 1/e + o(1). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

The Isomorphism Problem for Universal Enveloping Algebras of Lie Algebras

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ≅ U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H. Presented by D. Passman     

The quotient field as a torsion-free covering module

R will denote a commutative integral domain with quotient fieldQ. A torsion-free cover of a moduleM is a torsion-free moduleF and anR-epimorphism σ:F→M such that given any torsion-free moduleG and λ∈Hom _R (G, M) there exists μ∈Hom _R (G,F) such that σμ=λ. It is known that ifM is a maximal ideal ofR, R→R/M is a torsion-free cover if and only ifR is a maximal valuation ring. LetE denote the injective hull ofR/M thenR→R/M extends to a homomorphismQ→E. We give necessary and sufficient conditions forQ→E to be a torsion-free cover.