A Banach space E is said to have Mazur's property if every weak* sequentially continuous functional in E” is weak* continuous, i.e. belongs to E. Such spaces were investigated in [5] and [9] where they were called d-complete and μB-spaces respectively. The class of Banach spaces with Mazur's property includes the WCG spaces and, more generally, the Banach spaces with weak* angelic dual balls [4, p. 564]. Also, it is easy to see that Mazur's property is inherited by closed subspaces. The main goal of this paper is to present generalizations of some results of [5] concerning the stability of Mazur's property with respect to forming some tensor products of Banach spaces. In particular, we show in Sections 2 and 3 that the spaces E ⊗εF and Lp(μ, E) inherit Mazur's property from E andF under some conditions. In Section 4, we will also show the stability of Mazur's property under the formation of Schauder decompositions and some unconditional sums of Banach spaces.

Let (S, ≤) be a poset (partially ordered set), A(S) = Aut(S, ≤) its automorphism group and G ⊆ A(S) a subgroup. In the literature, various authors have studied sufficient conditions on G and the structure of (S, ≤) which imply that G is simple or perfect. Let us call (S, ≤) doubly homogeneous if each isomorphism between two 2-subsets of 5 extends to an isomorphism of (S, ≤). Higman [8] proved that if (S, ≤) is a doubly homogeneous chain then B(S), the group of all automorphisms of (S, ≤) with bounded support, is simple, and each element of B(S) is a commutator in B(S). Droste, Holland and Macpherson [5] showed that if (S, ≤) is a doubly homogeneous tree then its automorphism group again contains a unique simple normal subgroup in which each element is a commutator. Dlab [3] established similar results for various groups of locally linear automorphisms of the reals. Further results in this direction are contained in Glass [7]. It is the aim of this note to establish a common generalization and sharpening of the previously mentioned results.

All graphs considered in this article are finite connected, without loops and multiple edges. Let G be a graph and x be a vertex. The vertex neighbourhood graph (or υ-neighbourhood) of x in G (denoted by is the subgraph of G induced by the set of all vertices of G adjacent to x Analogously if f = xy is any edge of G, the edge neighbourhood graph (or e-neighbourhood) of f in G is the subgraph of G (denoted or induced by the set of all vertices of G which are adjacent to at least one vertex of the pair x, y and are different from x, y.

Here and throughout, A is a closed subalgebra of H∞ that contains the disk algebra and M(A) denotes the maximal ideal space of A. Because A contains the function fo(z) = z, we can define the fiber Mλ(A) of M(A) for λ ε ∂D (the unit circle) in the usual way; i.e., Mλ(A) = {φ ∈ M(A): fo(φ) = λ}. The Bergman space of the unit disk D is the L2(D, dx dy)-closure of A. Let be the orthogonal projection. For f ∈ L∞(D, dx dy), define the multiplication operator Mf: L2(D, dx dy)→ L2, (D, dx dy) by

and define the Toeplitz operator by

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.

Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

The methods of Lie theory have found widespread application in the study of the Lie algebras of vector fields on manifolds that arise naturally in geometric control theory (for some such applications, see [1]). Control systems on Lie groups themselves also have received considerable attention (see, for example, [9]). After reviewing basic facts about control systems on Lie groups, we derive the close relationship between attainable sets and Rådström's theory [12] of one-parameter semigroups of sets (Section 2). These ideas are then linked to the recently emerging Lie theory of semigroups [5]. The authors are indebted to the referee for pointing out some of the pertinent literature and analogous results from the area of geometric control.

A compact bordered Klein surface X of algebraic genus g ≥ 2 has maximal symmetry [6] if its automorphism group A(X) is of order 12(g — 1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ([6] and [9] among others).

We will denote the dimension of a subspace M of X by dim M and the codimension of M with respect to X by codxM or simply cod M if there is no danger of confusion. The classes of infinite dimensional and closed infinite codimensional subspaces of X will be denoted by and respectively with ℱ(X) and ℱ(X) denoting the classes of finite dimensional and of finite codimensional subspaces of X respectively. For a subspace M of X we denote the injection of M into X by and the quotient map from X onto the quotient space X/M by . Where there is no danger of confusion we will write JM and QM. The injection of X into its completion will be denoted by Jx. Letting X′ denote the continuous dual of X we remark that since X′ is isometric to ()′, these two spaces will be considered identical where convenient. The orthogonal complements of subsets M ⊂ X in X′ and K ⊂ X′ in X will be denoted by M⊥ and ⊥K respectively; M⊥X and X⊥K will be used if there is danger of confusion.

Since the pioneering work of W. G. Bade [3, 4] a great deal of work has been done on bounded Boolean algebras of projections on a Banach space ([11, XVII.3.XVIII.3], [21, V.3], [16], [6], [12], [13], [14], ]17], [18], [23], [24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(K)-module and x ε X, then a Bade functional of x with respect to C(K) is a continuous linear functional α on X such that, for each a in C(K) with a ≥ 0, we have

(i) α (ax) ≥0,

(ii) if α (ax) = 0, then ax = 0.

In [5] we exhibited the construction of faithful irreducible matrix representations of p-groups E and constructed their extensions to a semidirect product E. H, in case E and H satisfied suitable conditions. One of the major conditions was that the prime p had to be odd.

In this paper we assume the same conditions as in [5], but now with p = 2, in order to see if similar results can be obtained. Henceforth we will work with the following hypothesis.

In the sequel, given k, n ∈ ℕ, p ∈ [1, ∞] and a compact real interval I, we denote by Wk, p(I, ℝn) (simply by Wk,p(I if n = 1) the space of all functions u ∈ Ck−1(I, ℝn) such that uk−1 is absolutely continuous in I and u(k) ∈ Lp (I, ℝn).

Very recently, in [11], J. R. L. Webb and S. C. Welsh obtained the following existence result.

An almost Hermitian manifold (, J, g) with Riemannian connection is called nearly Kaehlerian if (xJ)X = 0 for any . The typical example is the sphere S6. The nearly Kaehlerian structure J for S6 is constructed in a natural way by making use of Cayley division algebra [3]. It is because of this nearly Kaehler, non-Kaehler, structure that S6 has attracted attention. Different classes of submanifolds of S6 have been considered by A. Gray [4], K. Sekigawa [5] and N. Ejiri [2]. In this paper we study 2-dimensional totally real submanifolds of S6. These are submanifolds with the property that for every x є M, J (Tx M) belongs to the normal bundle v. For this class we have obtained the following result.

Let Σg denote a compact orientable surface of genus g ≥ 2. We consider finite groups G acting effectively on Σg and preserving the orientation—for short, G acts on Σg or Gis a symmetry group of Σg. Each surface Σg admits only finitely many symmetry groups G and the orders of these groups are bounded by Wiman's bound of 84(g – 1). This bound is attained for infinitely many values of g [12], see also [9], and all values of g ≤ 104 for which it is attained are known [4].

Following Masser and Shiu [6] we say that a positive integer n is sparsely totient if

Here φ is the familiar Euler totient function. We write ℱ for the set of sparsely totient numbers. In [6] several results are proved about the multiplicative structure of ℱ. If we write P(n) for the largest prime factor of n then it was shown (Theorem 2 of [6]) that

and infinitely often

Let Ω: be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of continuous E-valued functions on Ω under supnorm. It is well known [3, p. 182] that if F is a Banach space then any bounded linear operator T:C(Ω, E)→ F has a finitely additive vector measure G defined on the σ-field of Borel subsets of Ω with values in the space ℒ(E, F**) of bounded linear operators from E to the second dual F** of F. The measure G is said to represent T. The purpose of this note is to study the interplay between certain properties of the operator T and properties of the representing measure G. Precisely, one of our goals is to study when one can characterize nuclear operators in terms of their representing measures. This is of course motivated by a well-known theorem of L. Schwartz [5] (see also [3, p. 173]) concerning nuclear operators on spaces C(Ω) of continuous scalar-valued functions. The study of nuclear operators on spaces C(Ω, E) of continuous vector-valued functions was initiated in [1], where the author extended Schwartz's result in case E* has the Radon-Nikodym property. In this paper, we will show that the condition on E* to have the Radon-Nikodym property is necessary to have a Schwartz's type theorem. This leads to a new characterization of dual spaces E* with the Radon-Nikodym property. In [2], it was shown that if T:C(Ω, E)→ F is nuclear than its representing measure G takes its values in the space (E, F) of nuclear operators from E to F. One of the results of this paper is that if T:C(Ω, E)→ F is nuclear then its representing measure G is countably additive and of bounded variation as a vector measure taking its values in (E, F) equipped with the nuclear norm. Finally, we show by easy examples that the above mentioned conditions on the representing measure G do not characterize nuclear operators on C(Ω, E) spaces, and we also look at cases where nuclear operators are indeed characterized by the above two conditions. For all undefined notions and terminologies, we refer the reader to [3].

A well-known theorem of E. Posner [10] states that if the composition d1d2 of derivations d1d2 of a prime ring A of characteristic not 2 is a derivation, then either d1 = 0 or d2 = 0. A number of authors have generalized this theorem in several ways (see e.g. [1], [2], and [5], where further references can be found). Under stronger assumptions when A is the algebra of all bounded linear operators on a Banach space (resp. Hilbert space), Posner's theorem was reproved in [3] (resp. [12]). Recently, M. Mathieu [8] extended Posner's theorem to arbitrary C*-algebras.

The purpose of this paper is to establish some basic properties of the Wielandt subgroup of a polycyclic group. The Wielandt subgroup of a group G is defined to be the intersection of the normalisers of all the subnormal subgroups of G and is denoted by ω(G). In 1958 Wielandt [9] showed that any minimal normal subgroup with the minimum condition on subnormal subgroups is contained in the Wielandt subgroup: it follows that the Wielandt subgroup of an artinian group is nontrivial. In contrast, the Wielandt subgroup of a polycyclic group can be trivial; an easy example is given by the infinite dihedral group. We will show that the Wielandt subgroup of a polycyclic group is close to being central.

Much of the research done by different authors on the lattice of kernel functors (equivalently, linear topologies) has been summarized by Golan in [2]. More recently, the rings whose lattices of kernel functors are linearly ordered were introduced in [3] as a categorical generalization of valuation rings in the non-commutative case. Results (and examples) in [3] show that there is an abundance of non-commutative rings R whose lattices (R), both in Mod-R and R-Mod, are simultaneously linearly ordered; however, the question of the symmetry of this condition remained open. Here we will prove that, for every natural number n≥3, there exists a ring Rn such that (Mod-Rn) is a linearly ordered lattice of n elements, whereas (Rn-Mod) is not linearly ordered.

If A is a semiprime Banach algebra, soc A, rad A the socle and radical of A, then Soc A ∩ rad A = (0). This elementary result enables us to prove some results concerning algebraic ideal and algebraic elements modulo the socle of A. We also deduce several conditions for A equivalent to the condition dim A <+∞.

A longstanding open question in low dimensional topology was raised by J. H. C. Whitehead in 1941 [9]: “Is any subcomplex of an aspherical, two-dimensional complex itself aspherical?” The asphericity of classical knot complements [7] provides evidence that the answer to Whitehead's question might be “yes”. Indeed, each classical knot complement has the homotopy type of a two-complex which can be embedded in a finite contractible two-complex. This property is shared by a large class of four-manifolds; these are the ribbon disc complements, whose asphericity has been conjectured, and even claimed, but never proven. (See [4] for a discussion.) It is reasonable and convenient to formulate the following.