Let G be a locally compact group and let Ĝ be its character group. Among other results, the minimal left ideals of L∞(G)* and LUC(G)* are all found when G is abelian. The main tool used for this study is the set of (topological) χ-invariant functionals (χ∈Ĝ), defined in this paper.

As is known, on a locally compact group G, the mere assumption of pointwise convergence of a sequence (n) of continuous positive definite functions implies uniform convergence of (n) to on compact subsets of G. This result was first proved in 1947 by Raikov8 (and independently by Yoshizawa9). An interesting discussion of the relationship between such theorems and various Cramr-Lvy theorems of the 1920s and 1930s, concerning the Central Limit Problem of probability, is given by McKennon(7, p. 62).

Suppose that A is a Banach algebra and let A be the second dual algebra of A equipped with the first Arens product 3. In this paper we characterize compact and weakly compact multipliers of A, when A possesses a bounded approximate identity and is a two sided ideal in A. We use this to study the isomorphisms between second duals of various classes of Banach algebras satisfying the above properties.

Let n be a natural number, with n ≥ 2. Let Kn denote the set of θ in Euclidean space Rn for which θ1, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y,…. We write

Inner product is denoted by θø. In Rl, ‖θ‖ denotes distance to the nearest integer.

Let X, Y be two Banach spaces. By L(X, Y) (resp. K(X, Y)) we denote the Banach space of all bounded, linear (resp. compact, bounded, linear) operators from X into Y. Several papers have been devoted to the question of when c0 embeds isomorphically into K(X, Y) (see 5, 8, 9 and their references) and its relationship with the following question:

(i) is K(X, Y) always uncomplemented in L(X, Y) when L(X, Y)K(X, Y)?

A subspace X of a Banach space Y is called an M-ideal if there is an L-projection P on Y* whose kernel is X1, the annihilator of X; that is, we have

In the case that Y is the bidual of X, a natural linear projection on X*** = Y* with kernel X1 is available, namely .

In this paper, we establish conditions for a continuous family of circle maps to have a continuous family of invariant measures in the appropriate topologies, and demonstrate the necessity of some of the conditions.

A family {S1, ,Sk} of contracting affine transformations on Rn defines a unique non-empty compact set F satisfying . We obtain estimates for the Hausdorff and box-counting dimensions of such sets, and in particular derive an exact expression for the box-counting dimension in certain cases. These estimates are given in terms of the singular value functions of affine transformations associated with the Si. This paper is a sequel to 4, which presented a formula for the dimensions that was valid in almost all cases.

An amenable semigroup of positive linear unital mappings on a W*-algebra is considered. Two main questions are dealt with: the existence of a normal faithful state invariant with respect to this semigroup and the description of ergodicity conditions. An explicit form of the ergodic projection, useful in treating the ergodicity problems, is also derived.

We are studying the integral inequality

where all functions appearing are defined and increasing on the right half-axis and take the value zero at zero. We are interested in determining when the inequality admits solutions u(x) which are non-vanishing in a neighbourhood of zero. It is well-known that if ψ(x) is the identity function then no such solution exists. This due to the fact that the operator defined by the integral on the right-hand side of the equation is linear and compact. So if we are interested in non-trivial solutions it is natural to require that ψ(x) > 0 at least for all non-zero points in some neighbourhood of zero. One of the typical examples is the power function ψ(x) = xα, where α < 1. This situation was explored in [2]. The functions a(x) that admit non-zero solutions were characterized by Bushell in [1]. For a general approach to the problem we refer to [2], [3] and [4].

Let E be a separable symmetric sequence space, and let CE be the unitary matrix space associated with E, i.e. the Banach space of all compact operators x on l2 so that s(x) E, with the norm , where are the s-numbers of x. One of the interesting subjects in the theory of the unitary matrix spaces is the clarification of correlation between the geometric properties of the spaces E and CE. A series of results in this direction related with the notions of type, cotype and uniform convexity of the spaces CE has been already obtained (see 13).

A systematic analysis of standard systems and modular systems for which one can develop TomitaTakesaki theory for algebras of unbounded operators is presented. Such systems arise in the Wightman quantum field theory. The connection between such systems and the theory of local nets of von Neumann algebras initiated by Araki and HaagKastler is discussed.

Certain interesting distributions are characterized by two or more symmetry properties, and it is a challenge to link these properties in an effective way. Dynkin's advice Discuss the simplest case first is always the best, and here we follow it (and avoid technical jargon) by discussing in detail just one concrete example, which concerns a discounted version of the arc-sine law. Theorem 1 B gives the symmetry properties for this case, and Theorem 1 A shows how they may be exploited to obtain an asymptotic result which we find rather surprising. Section 5 points to one of the generalizations which we shall examine in a sequel, and describes the natural formulation of the strange symmetry property which we first meet at (16b). (The sequel will be in many ways very different from this paper, but will share with it the fact that its results are not those which we originally conjectured.) In Sections 6 and 7 we consider some of the bizarre numerical analysis associated with the current problem.

In this paper we shall prove an existence theorem and give applications of an outgoing solution of the following problem:

where L(x, x) is a second order elliptic differential operator with a potential term q(x), is an exterior domain of ℝn (where n 2) with the C2-class boundary , k is an element of the complex plane or of a logarithmic Riemann surface, and B is either a Dirichlet boundary condition or of the form Bu = vj(x) ajk(x) ku + (x)u with the unit outer normal vector v(x) = (vl,, vn) at x.