Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that for any group element other than the identity there is a representation sending it into a matrix other than the unit matrix. If G is commutative, the representations are necessarily one-dimensional. It is an immediate consequence of the Peter-Weyl theorem that the converse also holds: if every representation is one-dimensional, G is commutative. The main theorem in the present paper is a generalization of this result to the case where the representations have bounded degree.

This paper is the first in a series dealing with Banach spaces L whose elements are functions on a measure space S. If W is a family of non-negative weight functions wα we sometimes write LWp when the norm is given as

A linear algebra is called a Jordan algebra if it satisfies the identities

(1) ab = ba, (a2b) a = a2(ba).

It is well known that a linear algebra S over a field of characteristic different from two is a Jordan algebra if there is an isomorphism a → a of the vector-space underlying S into the vector-space of some associative algebra A such that

1,

where the dot denotes the multiplication in A. Such an algebra S is called a special Jordan algebra.

A finite net N of degree k, order n, is a geometrical object of which the precise definition will be given in §1. The geometrical language of the paper proves convenient, but other terminologies are perhaps more familiar. A finite affine (or Euclidean) plane with n points on each line is simply a net of degree n+ 1, order n (Marshall Hall [1]). A loop of order n is essentially a net of degree 3, order n (Baer [1], Bates [1]). More generally, for , a set of k —2 mutually orthogonal n ⨯ n latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence (Bose [1]) between affine planes and complete sets of orthogonal latin squares.

The introduction of a mathematics student to formal integration theory usually follows the lines laid down by Riemann and Darboux. Later a change of ideas is necessary if he tackles Lebesgue's more powerful theory, and connections between the two are laboriously constructed. On the other hand, the commonest method of evaluating an integral is through the operation inverse to differentiation (the indefinite integral taken between limits). We refer to this as the calculus integral; few realize that this can succeed in cases where even the Lebesgue integral does not exist, let alone the Riemann one. An example is given later.

Our objective in this sequel to $[18]$ is to develop extensions, to representations of tensor algebras over ${{C}^{*}}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling’s theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras.

1. Krall and Frink [2] have recently considered in connection with certain solutions of the wave equation a system of polynomials yn(x), {n = 0, 1, 2, …), where yn is defined as that polynomial solution of the differential equation

An ordinary graph is a finite linear graph which contains no loops or multiple edges, and in which all edges are undirected. In such a graph G, let N, L, and T denote respectively the number of nodes, edges, and triangles. One problem, suggested by P. Erdös (1), is to determine the minimum number of triangles when the number of edges is specified, subject to suitable restrictions.

Let Aj and Bj (1 ≦ j ≦ m) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by

(1)

We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the form

For example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.

The aim of this paper is two-fold: first, to extend the results of (4) to the exceptional finite Lie groups recently discovered by Chevalley (1), and, secondly, to give a construction which works simultaneously for the groups An, Bn, Cn, Dn, En, F4 and G2 (in the usual Lie group notation), and which depends only on intrinsic structural properties of these groups.

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equations

Birkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

A class of problems in the geometry of numbers, for which there are but fragmentary results, may be expressed in general terms, as follows. Let 5 be a given point-set and let G be a given discrete point-set, both in Euclidean n-space. Suppose that Λ is a lattice which contains G but no point of S not in G. Such lattices, if they exist, will be said to be admissible for S with respect to G, and the general problem is to investigate their properties and, if possible, classify them.

Let G be a group, ZG the group ring of G over the ring Z of integers, and U(ZG) the group of units of ZG. One method of investigating U(ZG) is to choose some property of groups and try to determine the groups G such that U(ZG) enjoys that property. For example Sehgal and Zassenhaus [9] have given necessary and sufficient conditions for U(ZG) to be nilpotent (see also [7]), and the same authors have investigated when U(ZG) is an FC (finite-conjugate) group [10]. For a survey of related questions, see [3]. In this paper we consider when U(ZG) contains a free subgroup of rank 2. We conjecture that if this does not happen, then every finite subgroup of G is normal, from which various other conclusions then follow (see Lemma 4).

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1, and λ = t — 1. A block design is completely characterized by its so-called incidence matrix. The existence of a block design with parameters v, k, and λ such that the corresponding incidence matrix is cyclic was shown in (3) to be equivalent to the existence of a cyclic difference set with parameters v, k, and λ. For certain values of the parameters, Hadamard matrices, block designs, and difference sets do coexist.

The concept of an s-ply transitive (1 ≤ s ≤ n) permutation group on n symbols is of considerable importance in the classical theory of finite permutation groups, which was in the height of its development in the period around the turn of the century. The obvious generalization to a permutation group which is s set-transitive (i.e., a group which, for each pair of s-element unordered subsets S, T of the given n symbols, contains a permutation which carries S into T) seems to have received little attention.

1. Introduction. A space of orderings will refer to what was called a “set of quasi-orderings” in [5]. That is, a space of orderings is a pair (X, G) where G is an elementary 2-group (i.e. x2 = 1 for all x ∈ G) with a distinguished element – 1 ∈ G, and X is a subset of the character group x(G) = Horn (G, {1, –1};) satisfying the following properties:

01: X is a closed subset of χ(G).

02: σ(−l) = −1 holds for all σ ∊ X.

03: X⊥ = {a ∊ G|χa = 1 for all a ∊ X} = 1.

04: If f and g are forms over G and if x ∊ Df⊗g, then there exist y ∊ Df and z ∊ Dg such that x ∊ D(y, z).

Recently, M. Hofmeister [4] counted all nonisomorphic double coverings of a graph by using its Ζ2 cohomology groups, and J. Kwak and J. Lee [5] did the same work for some finite-fold coverings. In this paper, we give an algebraic characterization of isomorphic graph bundles, from which we get a formula to count all nonisomorphic graph-bundles. Some applications to wheels are also discussed.

Let ƒ:H → (—∞,∞] be lower semicontinuous, where H is a real Hilbert space. An approach based upon nonsmooth analysis and optimization is used in order to characterize monotonicity of ƒ with respect to a cone, as well as Lipschitz behavior and constancy. The results, which involve hypotheses on the proximal subgradient ∂ πƒ, specialize on the real line to yield classical characterizations of these properties in terms of the Dini derivate. They also give new extensions of these results to the multidimensional case. A new proof of a known characterization of convexity in terms of proximal subgradient monotonicity is also given.

Soient E un espace compact (par exemple l' intervalle et μ une mesure de Radon sur E. Une fonction numérique f intégrable pour μ définit une mesure sur l'ensemble des parties mesurables (pour μ) de E. On peut aussi considérer la forme linéaire qu'elle définit sur l'espace C des fonctions numériques continues dans E, et cette forme linéaire est continue pour la topologie définie par la norme . Si on appelle encore “mesure” une forme linéaire continue sur C, le théoréme de Lebesgue-Nikodym classique caractérise celles de ces “mesures” forme qui sont de la . par une condition de “continuité absolue” par rapport à μ.

The q-shifted factorial (a)n or (a; q)n is

and an empty product is interpreted as 1. Recently, Askey and Wilson [6] introduced the polynomials

1.1

where

1.2

and

1.3

We shall refer to these polynomials as the Askey-Wilson polynomials or the orthogonal 4ϕ3 polynomials. They generalize the 6 — j symbols and are the most general classical orthogonal polynomials, [2].