If R is a ring with identity and E is the semigroup of identity preserving ring endomorphisms of R then the fixed ring of E is denned by for each φɛE}. R is said to be rigid if RE = R.Equivalently, R is rigid if and only if

In a recent paper (13), we introduced the class of strongly E-reflexive inversesemigroups. This class was shown to coincide with the class of those inverse semigroups which are semilattices of E-unitary inverse semigroups. In particular, therefore, E-unitary inverse semigroups and semilattices of groups are strongly E-reflexive, and in fact so are subdirect products of these two types of semigroups.

We recall that a JC-algebra (Størmer (3)) is a norm closed Jordan algebra of self-adjoint operators on a Hilbert space. Recently, Alfsen, Shultz, and Størmer (1) have introduced a class of abstract normed Jordan algebras called JB-algebras, and have proved that every special JB-algebra is isometrically isomorphic to a JC-algebra. We show that this result brings to a satisfactory conclusion the discussion in (2) of certain wedges W in Banach algebras and their related Jordan algebras W–W, and leads to two characterisations of the bicontinuously isomorphic images of JC-algebras.

The Hardy-Littlewood-Pólya inequality in question can be written in the form

Here and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean that

defines a locally integrable function Kf for which (1.1) holds.

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if

(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

The study of radicals in general categories has followed several lines of development. The problem of defining radical properties in general categories has been considered by Kurosh and Shul'geifer, see (7). Under mild conditions on their categories they obtain sufficient conditions for the existence of radical functors which are closely related to radical properties. Another approach is by Maranda (5) and Dickson (3) who studied idempotent radical functors and torsion theories in abelian categories. Our aim has been to study radical functors in as general a category as possible. To this end we introduce the concept of an R-category. The categories of rings, modules, near-rings, groups and Jordan algebras are all examples of R-categories.

In this note the volumes of certain regions in the n-sphere will be found in two ways: (a) by using a symmetry argument, (b) by expressing the volumes as repeated integrals over the (n-l)-cube. By considering the 4 and 5 spheres and equating the integrals obtained by method (b) to the solution obtained by method (a) we evaluate integrals of the form

for certain values of a, b and c; it does not appear easy (if indeed it is possible) to evaluate these integrals by direct methods.

Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the form

It has been discovered independently by many authors that, in terms of this notation, the coefficient

In (5) the author showed how to construct all inverse semigroups from their trace and semilattice of idempotents: the construction is by means of a family of mappings between ℛ-classes of the semigroup which we refer to as the structure mappings of the semigroup. In (7) (see also (8) and (9)) K. S. S. Nambooripad has adopted a similar approach to the structure of regular semigroups: he shows how to construct regular semigroups from their trace and biordered set of idempotents by means of a family of mappings between ℛ-classes and between ℒ-classes of the semigroup which we again refer to as the structure mappings of the semigroup. In the present paper we aim to provide a simpler set of axioms characterising the structure mappings on a regular semigroup than the axioms (R1)-(R7) of Nambooripad (9). Two major differences occur between Nambooripad's approach (9) and the approach adopted here: first, we consider the set of idempotents of our semigroups to be equipped with a partial regular band structure (in the sense of Clifford (3)) rather than a biorder structure, and second, we shall enlarge the set of structure mappings used by Nambooripad.

In this paper we shall examine the relationship between the numerical range of aninner derivation, and that of its implementing element.

Much of this paper is taken from the author's doctoral thesis (5) written at theUniversity of Newcastle upon Tyne with the helpful guidance of Professor B. E.Johnson. The author also acknowledges the financial support of the Science ResearchCouncil during the period of this research.

In this note we shall consider the problem of uniquely continuing solutions of the parabolic equation

across an analytic arc σ: x=s1(t) satisfies the boundary data

We assume that u(x,t) is a classical solution of (1) in the domain D ={(x,t): s1(t)< x < s 2(t), 0 < t < t0}, continuously differentiate in D ∪ σ and define the “reflection” of D across σ by

In this paper we show that for each n∈{2,3,4,5,} the topological separationproperty Tn can be decomposed

where C, N2,…,Nn are purely lattice theoretic properties with the expected im-plications holding between them.

Let

be an entire function, where (αn) is a strictly increasing sequence of non-negativeintegers. The maximum modulus, M(r), the minimum modulus, m(r), and the maxi-mum term, μ(r), of f defined by

A largely untouched problem in the theory of inverse semigroups has been that of finding to what extent an inverse semigroup is determined by its lattice of inverse subsemigroups. In this paper we discover various properties preserved by lattice isomorphisms, and use these results to show that a free inverse semigroup ℱℐx is determined by its lattice of inverse subsemigroups, in the strong sense that every lattice isomorphism of ℱℐx upon an inverse semigroup T is induced by a unique isomorphism of ℱℐx upon T. (A similar result for free groups was proved by Sadovski (12) in 1941. An account of this may be found in Suzuki's monograph on the subject of subgroup lattices (14)).

A central trace on an order-unit Banach space A(K) is a centre-valued module homomorphism invariant under the group of symmetries of A(K).

The concept of central traces has been crucial in the theory of types for convex sets established in (4), (5). In von Neumann algebras, they are precisely the canonical centre-valued traces and their existence hinges on a fundamental theorem (Dixmier's approximation process) in von Neumann algebras. On the other hand, the existence of central traces in finite dimensional spaces is an easy consequence of Ryll-Nardzewski's fixed point theorem (5).

Let R be a ring and X a right R-module (all rings have identities and all modulesare unitary). The intersection of all non-zero submodules of X is denoted by μ(X). The module X is called monolithic if and only if μ(X)≠0 and in this case μ(X) is anessential simple submodule of X. (Recall that a submodule Y of X is essential if and only if Y ⊂ A ≠ 0 for every non-zero submodule A of X.) It is well known that a module X is monolithic if and only if there is a simple right R-module U such that X is a submodule of the injective hull E(U) of U. If x is a non-zero element of an arbitrary right. R-module X then by Zorn's Lemma there is a submodule Yx of X maximal with the property x ∉ Yx. It can easily be checked that X/Yx is monolithic and ⊂ Yx = 0, where the intersection is taken over all non-zero elements x of X.

Throughout this paper A denotes an operator function, holomorphic on a deletedneighborhood of a complex number λo, with values in the space ℒ(X,Y) of boundedlinear operators between two complex Banach spaces X and Y. In his survey article(7), I. C. Gohberg has defined for such an arbitrary operator function A the algebraic multiplicity RM(A;λo) and the reduced algebraic multiplicity RM(A;λo) of A at λo. In earlier papers (e.g., (8, 16)) these notions have been defined and studied for morerestricted classes of operator functions. In (8) Gohberg and Sigal treated the case when A is finite-meromorphic at λo, A(λ) is bijective for λ in some deleted neighbor-hood of λo and the constant term A0 in the Laurent expansion of A at λo is aFredholm operator. They proved that in this case

Let (S, ℳ) be a measurable space (that is, a set S in which is defined a σ-algebra ℳ of subsets) and X a locally convex space. A map M from ℳ to the family of all non-empty subsets of X is called a multimeasure iff for every sequence of disjoint sets An ɛ ℳ (n=1,2,… )with the series converges (in the sense of (6), p. 3) to M(A).

Strong summability has been studied by many authors, including Borwein and Cass (1, 2) who have studied sequence-to-sequence transforms. Here we studyintegral transforms; and due to the lack of a limitation theorem for such transforms, some results do not follow directly as in the sequence cases. The strong methods defined here can be applied to construct known and new strong summability methods. We do not give details here, but refer the reader to (3) with the suggestion that the natural scale operator method be used for Q and the named method for P. For example, with the Cesà methods, let P=(C,k,δ) and Q=(C,δ) to obtain .

In this paper we prove five structure theorems for groups with dihedral 3-normalisers. The interest in these theorems lies not so much in the results themselves as in what can be proved from them. The original versions of the results are contained in our doctoral thesis (1) where they are used to prove the following theorem, of which this paper, together with (2), (3) and other papers in preparation, will constitute a published proof: