This paper is concerned with the structure of the derivation algebra Der L of the Lie algebra L with split Cartan subalgebra H. The Fitting decomposition

of Der L with respect to ad ad H leads to a decomposition

where

This decomposition is studied in detail in Section 2, where the centralizer of ad L∞ in D0(H) is shown to be

which is Hom(L/L2, Center L) when H is Abelian. When the root-spaces La (a nonzero) are one-dimensional, this leads to the decomposition of Der L as

where T is any maximal torus of D0(H).

The results in this paper respond to two rather natural questions about a polar decomposition A = UP, where U is a unitary matrix and P is positive semidefinite. Let λ1, …, λn be the eigenvalues of A. The questions are:

• (A) When will |λ1|, …, |λn| be the eigenvalues of P?

• (B) When will λ1/|λ1|, …, λn/|λn| be the eigenvalues of U?

The complete answer to (A) is “if and only if U and P commute.” In an important special case the answer to (B) is “if and only if U2 and P commute.“

Since these matters are best couched in terms of two different inertias, we begin with a unifying definition of inertia which views all inertias from a single perspective.

For each square complex matrix A and each complex number z let m(A, z) denote the multiplicity of z as a root of the characteristic polynomial

Dans [3], Peter Hilton a introduit la notion de “localisation relative” d'une extension de groupes à noyau nilpotent. Etant donné une extension de groupes

oú N est nilpotent, il s'agit de l'introduire dans un diagramme commutatif:

oú NP est la P-localisation de N.

L'idée (issue de la Topologie algébrique) est de tourner autant que possible l'absence de localisation maniable dans la catégorie des groupes. J'ai moi-même exploité cette idée dans [6], dans le cas oú une action de groupe est exercée sur G.

This paper features strong and weak compactness in spaces of vector measures with relatively compact ranges in Banach spaces. Its tools are the measure-operator identification of [16] and [24] and the description of strong and weak compactness in spaces of compact operators in [10], [11], and [29].

Given a Banach space X and an algebra of sets, it is shown in [16] that under the usual identification via integration of X-valued bounded additive measures on with X-valued sup norm continuous linear operators on the space of -simple scalar functions, the strongly bounded, countably additive measures correspond exactly to those operators which are continuous for the coarser (locally convex) universal measure topology τ on . It is through the latter identification that the results on strong and weak compactness in [10], [11], and [29] can be applied to X-valued continuous linear operators on the generalized DF space to yield results on strong and weak compactness in spaces of vector measures.

We say that two elements e and f of a lattice are moderately separated provided e ∧ f = 0 and both (e′, f′) and (f′, e′) are modular pairs for all e′ ≦ e and f′ ≦ f. Here (e′, f′) a modular pair means that, for all g ≧ e′,

In the lattice of projections of a factor we show that e and f, with e ∧ f = 0, are modularly separated if and only if ‖(e – k)f‖ < 1 for some finite projection k ≦ e. From there we can show that a kind of “independence property” holds for modular separation in this case: if e and f are modularly separated and if e ∨ f and g are modularly separated, then e and f ∧ g are modularly separated.

The following well known inequality was first proved by Bernstein [2].

THEOREM A. If pn(x) is a polynomial of degree n, such that |pn(x)| ≦ 1 for –1 ≦ x = +1, then

1

The dominant n(1 – x2)–;1/2 is best possible only at the zeros of the Tchebychev polynomial

but the bound is precise at every interior point as far as the exponent of n is concerned.

Theorem A was extended to the case of higher derivatives by Duffin and Schaeffer in [4]. In that paper they make extensive use of the oscillation property of the polynomial Tn(x) and of the related function

In a recent paper [12], C. Mueller proved a general version of the functional LIL which unifies Strassen's LIL and the Lévy modulus of continuity for Brownian motion W(t). His theorem also contains other known forms of the LIL.

For each t ≧ 0, let be a family of points in the first quadrant of the plane. Let r ≦ 0; to each point (s0, l0), we associate a rectangle

Define Ar(t) to be the area of the union of these rectangles up to time t under the measure . Then, Theorem 1 [12, p. 166] states that for an increasing function h such that

the set of limit points of

in C[0, 1] is the closed unit ball of the reproducing kernel Hilbert space (rkhs) associated with Wiener measure.

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the set

We say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

In 1930, M. H. A. Newman proved a rather remarkable theorem which has become one of the classical theorems in topology. It has many important applications. A special case of Newman's Theorem is that a periodic homeomorphism of period n > 1 of a sphere S onto itself must have some orbit which is not contained in a “cap” smaller than a hemisphere. The general theorem is as follows:

THEOREM ([15]). Suppose that Mn is a connected (metric) n-manifold, U is a domain in Mn, andp is an integer greater than 1. Then there is a positive number d such that no uniformly continuous homeomorphism h of Mn onto itself of period p moves every point of U a distance < d. That is, there is x ∊ U so that the orbit of x under h has diameter ≦ d.

It is an old conjecture by P. Bankston that the category CompHaus of compact Hausdorff spaces and their continuous maps is not dually equivalent to any elementary P-class of finitary algebras (taken as a category with all homomorphisms between its members as maps), where elementary means defined by first order axioms, and a P-class is one closed under arbitrary (cartesian) products. One motivation for this conjecture is the fact that such a dual equivalence would make ultracopowers of compact Hausdorff spaces correspond to ultrapowers of finitary algebras, and one might expect this to have contradictory consequences.

As a possible step towards proving his conjecture, Bankston [2] showed that no elementary SP-class of finitary algebras can be dually equivalent to CompHaus. However, it was subsequently proved in [1] that the same holds for any SP-class of finitary algebras, using an argument independent of ultrapowers.

The purpose of this paper is to continue the investigation into Massey products defined on two dimensional polyhedra, initiated in [13]. It will be shown that for many such spaces there is a hyperbolic model which can be used to study Massey products. More precisely, Massey products may be interpreted as intersections of geodesies in the Poincaré model. These elements are called minimal Massey products and are the analogue of Massey products over a system considered in Porter's paper. They enjoy the property of being uniquely defined (without indeterminacy) and of being multilinear and natural. Minimal products also satisfy symmetry properties generalising the symmetry properties enjoyed by cup products.

A device which will be useful in the proof of the main theorem, 7.4, is the introduction of a class of complexes called basic complexes. These generalise the notion of a surface and each one houses a standard copy of a Massey product.