A group G is said to have the subnormal join property (s.j.p.) if the join of two (and hence of finitely many) subnormal subgroups of G is always subnormal in G. Following Robinson [6], we shall denote the class of groups having this property by . A particular subclass of is , consisting of those groups G in which the join of two subnormals is again subnormal in G and has defect bounded in terms of the defects of the constituent subgroups (for a more precise definition see Section 7 of [6]).

In [16], Wielandt showed that groups which satisfy the maximal condition for subnormal subgroups have the s.j.p. Many further results on groups with the s.j.p. were proved in [6] and [7]. In Sections 2 and 3 of this paper, it will be shown that several of these results can be exhibited as corollaries of a few rather more general theorems on the classes , .

One aspect of the study of 3-manifolds is to determine what finite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action? In the case of periodic transformations of homology spheres, Smith [18] classified the types of fixed point sets which could occur. For homology 3-spheres the fixed point set will be ∅, S0, S1, or S2. Fox [4] looked at periodic transformations of the three sphere which leave a knot invariant and, using Smith's classification of fixed point sets, determined that there were eight types of transformations according to how the fixed point set met the knot. For convenience we shall say a knot is (a, b)-periodic if there is a periodic transformation of S3 leaving the knot invariant with fixed point set homeomorphic to a and with the fixed point set meeting the knot in a set homeomorphic to b.

The aim of this paper is to extend to a suitable class of topological semigroups parts of well-defined theory of representations of topological groups. In attempting to obtain these results it was soon realized that no general theory was likely to be obtainable for all locally compact semigroups. The reason for this is the absence of any analogue of the group algebra Ll(G). So the theory in this paper is restricted to a certain family of topological semigroups. In this account we shall only give the details of those parts of proofs which depart from the standard proofs of analogous theorems for groups.

On a locally compact semigroup S the algebra of all μ ∊ M(S) for which the mapping and of S to M(S) (where denotes the point mass at x) are continuous when M(S) has the weak topology was first studied in the sequence of papers [1, 2, 3] by A. C. and J. W. Baker.

Let S denote the class of functions of the form

that are analytic and univalent in the unit disk Δ = {z:|z| < 1}, with S*(α) and K(α) designating the subclasses of S that are, respectively, starlike of order a and convex of order α, 0 ≦ α < 1. If f(z) and g(z) are analytic in Δ, we say that f(z) is subordinate to g(z), written f ≺ g, if there exists a Schwarz function w(z), w(0) = 0 and |w(z)| < 1 in Δ, such that f(z) = g(w(z)). A function f(z) = z + … is said to be in S*[A, B] if

(1)

and in K[A, B] if

(2)

For 1 ≦ p < ∞ let Hp denote the familiar Hardy space of analytic functions on the open unit disc D and let ‖·‖ denote the Hp norm. Let Sp denote the space of analytic functions f on D such that f′ ∊ Hp. In this paper we will describe the linear isometries of Sp into itself when Sp is equipped with either of two norms. The first norm we consider is given by

(1)

and the second by

(2)

(It is well known [1, Theorem 3.11] that f′ ∊ Hp implies continuity for f on D, the closure of D. Thus (2) actually defines a norm on Sp.) In the former case, with the norm defined by (1), we will show that an isometry of Sp induces, in a sense to be made precise in Section 2, an isometry of Hp and that Forelli's characterization [2] of the isometries of Hp can thus be used to describe the isometries of Hp.

Dans cette note, nous donnons une réponse à un problème posé dans [5]: nous démontrerons que le niveau de quasi-unipotence de la monodromie avec coefficients est le meilleur possible (voir le paragraphe 2 pour l'énoncé précis).

1. Faisceaux quasi-unipotents. Nous rappelons ici la définition et les propriétés fondamentales de ces faisceaux (voir [3], cf. [5]).

Par D nous désignons un disque ouvert de C centré à l'origine et de rayon aussi petit qu'on veut. Un faisceau V lisse (ou, suivant une autre terminologie, localement constant) constructible sur D* = D – {0} est dit quasi-unipotent (de niveau d) si la monodromie T opérant sur la fibre générique de V vérifie (Ta – 1)b = 0, où a et b sont des entiers non-négatifs (et b ≦ d).

Définition (1.1). Soient X un espace analytique, Y un sous-espace analytique de X, et V un faisceau constructible sur X – Y.

The classical Hankel transform of a conventional function ϕ on (0, ∞) defined formally by

was extended by Zemanian [21-23] to certain generalized functions of one dimension. Koh [9, 10] extended the work of [21] to n-dimensions, and that of [22] to arbitrary real values of μ. Motivated from the work of Gelfand and Shilov [6], Lee [11] introduced spaces of type Hμ and studied their Hankel transforms. The results of Lee [11] and Zemanian [21] are special cases of recent results obtained by the author and Pandey [14]. The aforesaid extensions are accomplished by using the so-called adjoint method of extending integral transforms to generalized functions. Dube and Pandey [2], Pathak and Pandey [15, 16] applied a more direct method, the so-called kernel method, for extending the Hankel and other related transforms.

Let E be a Banach space, A a subset of its dual E*.x0 ∊ A is said to be a w*Gδ point of A if there are xn ∊ E and scalars γn, n = 1,2, 3 … such that

Denote by w*Gδ{A} the set of all w*Gδ points of A. If S is a semigroup of maps on E* and K ⊂ E*, denote by

i.e., the set of points x* in the w*closure of K which are fixed points of S (i.e., sx* = x* for each s in S}. An operator will mean a bounded linear map on a Banach space and Co B will denote the convex hull of B ⊂ E.

In this paper, we investigate a conjecture of Dixmier [2] on the structure of basic cycles. Our interest in basic cycles arises primarily from the fact that the irreducible modules of a simple Lie algebra L having a weight space decomposition are completely determined by the irreducible modules of the cycle subalgebra of L. The basic cycles form a generating set for the cycle subalgebra.

First some notation: F denotes an algebraically closed field of characteristic 0, L a finite dimensional simple Lie algebra of rank n over F, H a fixed Cartan subalgebra, U(L) the universal enveloping algebra of L, C(L) the centralizer of H in U(L), Φ the set of nonzero roots in H*, the dual space of H, Δ = {α1, …, αn} a base of Φ, and Φ+ = {β1, …, βm} the positive roots corresponding to Δ.

Since the invention of the Pettis integral over forty years ago [11], the problem of recognizing the Pettis integrability of a function against an individual measure has been much studied [5, 6, 7, 8, 9, 20]. More recently, Riddle-Saab-Uhl [14] and Riddle-Saab [13] have considered the problem of when a function is integrable against every Radon measure on a fixed compact Hausdorff space. These papers give various sufficient conditions on a function that ensure this universal Pettis integrability. In this paper, we see how far these various conditions go toward characterizing universal Pettis integrability. We base our work on a w*-analogue of the core of a vector-valued function [8].

We also give some sufficient conditions that ensure that a Banach space has the so-called universal Pettis integral property (UPIP) and consider some particular examples of spaces with this property. It is interesting that in these examples some of the special set theoretic axioms that play an important role in the study of the stronger Pettis integral property [6, 7] make an appearance.

Let G be a compact Lie group and a an action of G on a C*-algebra as *-automorphisms. Let denote the set of G-finite elements for this action, i.e., the set of those such that the orbit {αg(x):g ∊ G} spans a finite dimensional space. is a common core for all the *-derivations generating one-parameter subgroups of the action α. Now let δ be a *-derivation with domain such that Let us pose the following two problems:

1. Is δ closable, and is the closure of δ the generator of a strongly continuous one-parameter group of *-automorphisms?

2. If is simple or prime, under what conditions does δ have a decomposition

where is the generator of a one-parameter subgroup of α(G) and is a bounded, or approximately bounded derivation?