Disjoint Elements Research Articles

The structure of a lattice-ordered group with a finite number of disjoint elements.

The structure of a lattice-ordered group with a finite number of disjoint elements.

Separation and Approximation in Topological Vector Lattices

Spectral theory in its lattice-theoretic setting proves abstractly that the indicators of measurable sets generate the space L of Lebesgue-integrable functions on an interval. We are concerned here with abstractions suggested by the fact that indicators of intervals suffice to generate L. Our results show that the approximation of arbitrary elements of a topological vector lattice rests upon the ability to separate disjoint elements/ and g by an operation that behaves in the limit like a projection annihilating/ and leaving g invariant.The introduction of this concept of separation together with the notion of limit unit leads (via the Fundamental Lemma) to abstract generalizations of the Radon-Nikodym Theorem (Theorem 1) and the Stone-Weierstrass Theorem (Theorem 3).

The Two-Sided Regular Representation of a Unimodular Locally Compact Group

We show that for any unimodular locally compact group G, the lattice of all closed linear subspaces in L2(G) which are invariant under right and left translations is a Boolean algebra. (L2(G) is the Hilbert space of functions on G squareintegrable relative to Haar measure.) This Boolean algebra B can be regarded as a kind of dual to the group, being in measure-theoretic respects at least, equivalent to the usual duals, in the cases for which a duality theory is already established (i.e. for abelian and compact groups). In particular, the decomposition of L2(G) relative to B can be looked on as a formal generalization of the Plancherel and Peter-Weyl theorems, in which integration over the character group or over the space of equivalence classes of irreducible representations, is generalized by integration over B. The orthogonality of coordinates of inequivalent irreducible representations has as a formal analog the fact that any two disjoint elements of B are orthogonal. The key result, on which these developments rest, asserts that the only bounded linear transformations on L2(G) which commute with all left translations are the right translations, and the (weak) limits of their linear combinations. In the case of a discrete group this theorem is known [2], but the proof for that case seems incapable of generalization, as it makes heavy use of the special nature of the group. Although the generalized Plancherel and Peter-Weyl theorems can be regarded as stating the decomposition of L2(G) under a one-sided regular representation of G (e.g. under the left regular representation, a -+ La for a e G, where (Laf) (x) = f(a-lx) for f e L2(G)), this viewpoint seems unlikely to lead to a unique analog to those theorems for arbitrary unimodular locally compact groups. The main difficulty which arises may be described briefly as follows. A complete reduction of a unitary representation of a group can be considered to amount to a choice of a maximal Boolean algebra of closed linear subspaces which are invariant under the group. Now in general, this Boolean algebra will not be unique, even within unitary equivalence. For example, in the case of the reduction of an infinite-dimensional Hilbert space under the action of the identity operator, there is one maximal Boolean algebra of (invariant) subspaces which is generated by its minimal elements, and another which contains no minimal elements. In the case of a one-sided regular representation of a group which is abelian or compact, the algebra happens to be unique (within unitary equivalence), for the following reasons: in the abelian case, the invariant subspaces under either onesided regular representation are obviously the same as those invariant under the two-sided (i.e. both) representations; in the compact case, the two-sided regular

A United Front

In an admirable artide on the Spanish situation by the Dominican Father H. Munoz these words occur: ‘What Spain needed was an intelligent organizer who could coordinate all the disjointed elements into a united front against the destructive forces of irreligion and anarchy.’ On reading this it is impossible not to reflect that this is an echo of what many English Catholics have been saying about their own country for a long time.The present position of Catholics in England is not so very far removed from that of Catholics in Spain just before the revolution. Indifference and lack of organization sold the pass and allowed the forces of an anti-religious Socialism to seize power. And not till the Catholics of Spain found their diurches pillaged, convents burned, schools secularized, religious orders expelled and a sheaf of anti-Catholic laws on the statute book did they wake up to the realities of the situation and organize themselves into a force which now promises to be the salvation of Spain.In England, a generation of ever-increasing indifference in matters of faith and morals has enabled the anti-God propaganda of Moscow to make tremendous progress. The traditional Conservative and Protestant party, product of the Reformation, exists in a fog. In matters of faith and morals they simply do not know where they stand, having neither standards nor guidance. And from not knowing to not caring is but a short step.Liberalism no longer counts. The more extreme Liberals have gone over to Labour. A few have thrown in their lot with the Conservatives. The rest plough a lonely furrow in the wilderness.