An algebra of bounded operators on a Hilbert space H is said to be reductive if it is unital, weakly closed and has the property that if M ⊂ H is a (closed) subspace invariant for every operator in , then so is M⊥. Loginov and Šul'man [6] and Rosenthal [9] proved that if is an abelian reductive algebra which commutes with a compact operator K having a dense range, then is a von Neumann algebra. Note that in this case every invariant subspace of is spanned by one-dimensional invariant subspaces. Indeed, the operator KK* commutes with . Hence its eigenspaces are invariant for , so that H is an orthogonal sum of the finite-dimensional invariant subspaces of From this our claim easily follows.

One of the fundamental results on graph coloring is the following classical theorem of Brooks.

BROOKS’ THEOREM. Suppose that k ≧ 3 and that G is a k-regular graph which does not induce a (k + 1)-clique. Then G is k-colorable.

Brooks proved his theorem in [1]; several more recent proofs have appeared in [3], [4] and [5]. All the proofs of this theorem have the common feature of applying only to finite graphs; the transition to infinite graphs can be accomplished by a very standard implementation of the Compactness Theorem (or some other equally noneffective device such as the theorem of deBruijn and Erdös [2] asserting that a graph is k-colorable if and only if each of its finite subgraphs is). Thus, it is not immediately apparent that an effective version of Brooks’ Theorem exists. It is our purpose to show, however, that the effective analogue of Brooks' Theorem is indeed true.

In this paper we study intersection properties of measurable sets with positive measure in a probability measure space, or equivalently, intersection properties of open subsets on a compact space with a strictly positive measure.

The first result in this direction is due to Erdös and it is a negative solution to the problem of calibers on such spaces. In particular, under C.H., Erdös proved that Stone's space of Lebesque measurable sets of [0, 1] modulo null sets, does not have ℵ1-caliber.

Lorsque Kn = Q(ω) est une extension algébrique de degré n sur Q telle que

avec D ∈ N, d ∈ Z, d|D2 et D2 + 4d > 0, nous avons prouvé [1] en utilisant certaines idées de Halter-Koch et Stender [2] que si

alors

est un système indépendant d'unités de Kn.

Arhangel'skiĭ proved around 1959 [1] that, for the class of perfectly normal locally compact spaces, metacompactness and paracompactness are equivalent. It is shown to be consistent that this equivalence holds for the (larger) class of normal locally compact spaces (answering a question of Tall [8], [9]).

The consistency of the existence of locally compact normal noncollectionwise Hausdorff spaces has been known since 1967. It is shown that the existence of such spaces is independent of the axioms of set theory, thus establishing that Bing's example G cannot be modified under ZFC to be locally compact.

All topological spaces are assumed to be Hausdorff.

First, a definition and three standard lemmata are needed.

Let V ≠ 0 be a vector space of dimension n over a finite field of order qm for a prime q. Of course, GL(n, qm) denotes the group of -linear transformations of V. With few exceptions, GL(n, qm) is non-solvable. How large can a solvable subgroup of GL(n, qm) be? The order of a Sylow-q-subgroup Q of GL(n, qm) is easily computed. But Q cannot act irreducibly nor completely reducibly on V.

Suppose that G is a solvable, completely reducible subgroup of GL(n, qm). Huppert ([9], Satz 13, Satz 14) bounds the order of a Sylow-q-subgroup of G, and Dixon ([5], Corollary 1) improves Huppert's bound. Here, we show that |G| ≦ q3nm = |V|3. In fact, we show that

where

This paper is a continuation of [5]. As stated there, the problem is to explicitly decompose the space L2 = L2(G(F)\G(A)) into simpler invariant subspaces, and to deal with the associated continuous spectrum in case G is a connected reductive algebraic group defined over a global function field. In that paper the solution was begun by studying Eisenstein series associated to cusp forms on Levi components of parabolic subgroups; these Eisenstein series and the associated intertwining operators were shown to be rational functions satisfying functional equations. To go further it is necessary to consider more general Eisenstein series and intertwining operators, and to show that they have similar properties. Such Eisenstein series arise from the cuspidal ones by a residue taking process, which is detailed in a disguised form suitable for induction in the first part of this paper: the main result is a preliminary form of the spectral decomposition.

The purpose of this paper is to study the operator Δ + q. Here Δ is the Laplace–Beltrami operator on a compact Lie group G and q is a matrix coefficient of a representation of G. We are able to calculate the powers of Δ + q acting on the function qku. This is done in Section 2 and the reader is refered there for definitions of the special functions q and u.

The interest in the operator Δ + q comes originally from physics and in particular from the Schrödinger equation. This is described in [4]. Here we are restricting ourselves to mathematical questions and shall not consider any applications to physics.

In this paper we take the heat equation with potential as

(1.1)

with , the upper half plane, and initial data f(x, 0) = qk(x)u(x).

An ovoid in an orthogonal vector space V of type Ω+(2n, q) or Ω(2n – 1, q) is a set Ω of qn–1 + 1 pairwise non-perpendicular singular points. Ovoids probably do not exist when n > 4 (cf. [12], [6]) and seem to be rare when n = 4. On the other hand, when n = 3 they correspond to affine translation planes of order q2, via the Klein correspondence between PG(3, q) and the Ω+(6, q) quadric.

In this paper we will describe examples having n = 3 or 4. Those with n = 4 arise from PG(2, q3), AG(2, q3), or the Ree groups. Since each example with n = 4 produces at least one with n = 3, we are led to new translation planes of order q2.

Let M be a w*-algebra (Von Neumann algebra), τ a semifinite, faithful, normal trace on M. There exists a w*-dense (i.e., dense in the σ(M, M*)-topology, where M* is the predual of M) *-ideal J of M such that τ is a linear functional on J, and

(where |x| = (x*x)1/2) is a norm on J. The completion of J in this norm is Lp(M, τ) (see [2], [8], [7], and [4]).

If M is abelian, in which case there exists a measure space (X, μ) such that M = L∞(X, μ), then Lp(X, τ) is isometric, in a natural way, to Lp(X, μ). A natural question to ask is whether this situation persists if M is non-abelian. In a previous paper [5] it was shown that it is not possible to have a linear mapping