In this article, we study some questions related to the complementation and the Hahn-Banach property for subspaces of lp, for 0 < p < 1. Some results which are stated here have appeared in the work of W. J. Stiles [4, 5] and N. Popa [3], but our proofs are simpler. We solve a problem raised by Popa [3], concerning complemented copies of lp contained in lp.

The following result in the theory of numerical ranges in Banach algebras is well known (see [3, Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w*-continuous states of . As a consequence of the results in this paper the following

Let Mn(ℂ) be the algebra of n × n complex matrices, and let be its unitary group. Given A, B ε Mn(ℂ), the A-numerical radius of B is the nonnegative quantity

In particular, for A = diag(1, 0, …, 0) it reduces to the classical numerical radius r(B) = max||x*Bx|:x*x = 1}. In [1] Goldberg and Straus proved that rA is a generalized matrix norm (i.e. a positive definite seminorm) on Mn(ℂ) if and only if A is nonscalar and tr A ≠ 0. This result agrees with the well-known fact that the classical numerical radius r is a generalized matrix norm. The nontrivial part of the proof is to show that if A is nonscalar and tr A ≠ 0 then rA is positive definite; that is, for any B ε Mn(ℂ), tr(AU*BU) = 0 for all U ε implies B = 0. The proof given in [1] is computational and involves the use of differentiation on matrices. Later Marcus and Sandy [2] gave three elementary proofs of the result. Their proofs are still computational in nature and two of them need knowledge of multilinear algebra.

The problem of describing the subsemigroup generated by the idempotents in various natural semigroups has received the attention of several semigroup theorists ([1], [2], [3], [5], [7]). However, in those cases where the parent semigroup is in fact the multiplicative semigroup of a natural ring, the known ring structure has not been exploited. When this ring structure is taken into account, proofs can often be streamlined and can lead to more general arguments (such as not requiring that the elements of the semigroup be already transformations of some known structure).

In his Habilitationsschrift [3] B. Fischer introduced the concept of a normally embedded subgroup of a finite group. A subgroup of a finite group G is said to be normally embedded in G if each of its Sylow subgroups is a Sylow subgroup of a normal subgroup of G. Meanwhile this concept has become of considerable importance in the theory of finite soluble groups and has been studied by various authors. However, in infinite group theory, normally embedded subgroups seem to have received little attention. The object of this note is to study normally embedded subgroups of locally soluble FC-groups.

The word problem for free bands (idempotent semigroups) was solved by Green and Rees [4] in an early paper. They also established certain properties of the free band. This was followed by McLean [6] who provided a general structure theory for bands with some indication as to the structure of the free band. Since then a great many papers have appeared dealing with various aspects of the topic of bands and their varieties. A different solution of the word problem for free bands was recently given by Siekmann and Szabó [9]. For a discussion of bands, see the books [5] and [8].

Several definitions of a semigroup of quotients have been proposed and studied by a number of authors. For a survey, the reader may consult Weinert's paper [8]. The motivation for many of these concepts comes from ring theory and the various notions of rings of quotients. We are concerned in this paper with an analogue of the classical ring of quotients, introduced by Fountain and Petrich in [3].

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B (X) (K(X)). Let σ(A) and σa(A) denote, respectively, the spectrum and approximate point spectrum of an element A of B(X). Set

σem(A)and σeb(A) are respectively Schechter's and Browder's essential spectrum of A ([16], [9]). σea (A) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([13], [14]). In this note we characterize σab(A) and show that if f is a function analytic in a neighborhood of σ(A), then σab(f(A)) = f(σab(A)). The relation between σa(A) and σeb(A), that is exhibited in this paper, resembles the relation between the σ(A) and the σeb(A), and it is reasonable to call σab(A) Browder's essential approximate point spectrum of A.

Braid groups were introduced by Artin [1]. These groups have been studied extensively—see [2], [9] and the references cited there. Recently work has been done on “circular” braid groups and other “braid-like” groups [7], [10]. In this paper we formulate the concept of a generalized braid group, and we begin a study of the structure of such groups. In particular for such a group G1, there is a homomorphism from G onto the infinite cyclic group, the kernel of which is the derived group G1 of G. We study G1. Our results generalize results of Gorin and Lin [5], who considered the case when G is a classical braid group B(n ≥ 3). They showed that is free abelian of rank 2 if n = 3, 4 and is trivial if n ≥ 5. They also showed that is finitely presented.

An algebra A factors if, for each a ∈ A, there exist b, c ∈ A with a = bc. A bounded approximate identity for a Banach algebra A is a net (eα) ⊂ A such that aeα → a and eαa → a for each a ∈ A and such that sup ‖eα ‖ < ∞. It is well known [2, 11.10] that if A has a bounded approximate identity, then A factors. But a Banach algebra may factor even if it does not have a bounded approximate identity: an example which is non-commutative and separable, and an example which is commutative and nonseparable, are given in [3, §22]. However, we do not know an example of a commutative, separable Banach algebra which factors, but which does not have a bounded approximate identity. See 4 for related work.

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].

For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

V-rings and their generalisations have been studied extensively in recent years [2], [3], [5],[6], [7]. All the rings we consider will be associative rings with 1 ≠ 0 and all the modules considered will be unitary left R-modules. All the concepts will be left-sided unless otherwise mentioned. Thus by an ideal in R we mean a left ideal of R. A ring R is called a V-ring (respectively a GV-ring) if every simple (resp. simple, singular) module is injective. An R-module M is called p-injective if any homomorphism f: I → M with I a principal left ideal of R can be extended to a homomorphism g: R → M. A ring R is called a p-V-ring (resp. a p-V'-ring) if every simple (resp. simple, singular) module over R is p-injective. The object of the present paper is to introduce torsion theoretic generalizations of p-V-rings and prove results similar to those obtained by Yue Chi Ming about p-V-rings and p-V'-rings [6], [7]. For any M ∈ R-mod, J(M) will denote the Jacobson radical of M and Z(M) the singular submodule of M. For any λ ∈ R, we denote the left annihilator { r ∈ R| rλ =0 } of λ in R by l(λ).

In a recent paper [3], D. W. Lewis proposed the following conjecture. (The notation is the same as that in [2] and [3].)

Conjecture. Let F be a field of characteristic not 2 and let a1, b1…, an, bn ∈ Fx. The tensor product of quaternion algebras

is a division algebra if and only if the quadratic form over F

is anisotropic.

This equivalence indeed holds for n = 1 as is well known [2, Theorem 2.7], and Albert [1] (see also [4, §15.7]) has shown that it also holds for n = 2. The aim of this note is to provide counterexamples to both of the implications for n ≥ 3.

Let M be a differentiable manifold of dimension m. A tensor field f of type (1, 1) on M is called a polynomial structure on M if it satisfies the equation:

where a1, a2, …, an are real numbers and I denotes the identity tensor of type (1, 1).

We shall suppose that for any x ∈ M

is the minimal polynomial of the endomorphism fx: TxM → TxM.

We shall call the triple (M, f, g) a polynomial Lorentz structure if f is a polynomial structure on M, g is a symmetric and nondegenerate tensor field of type (0, 2) of signature

such that g (fX, fY) = g(X, Y) for any vector fields X, Y tangent to M. The tensor field g is a (generalized) Lorentz metric.

Throughout this paper all rings are assumed commutative with identity. Among integral domains, Dedekind domains are characterized by the property that every ideal is a product of prime ideals. For a history and proof of this result the reader is referred to Cohen [2, pp. 31–32]. More generally, Mori [5] has shown that a ring has the property that every ideal is a product of prime ideals if and only if it is a finite direct product of Dedekind domains and special principal ideal rings (SPIRS). Rings with this property are called general Z.P.I.-rings.

On p. 244 of Glasgow Math. J.27 (1985) on the right hand side of one of the 6 equations characterizing the 4 fixed points of the involution v a sign error has occurred.The relevant equation should read

y0y3y5y6=–1,

or the points would not lie on the curve.

Correcting the error unfortunately invalidates the model of an elliptic curve given in §6, which therefore has to be re-evaluated. First we find, in the notation of the paper,

2 f (x) = ((r + 1)/(R + 2))2.