Let 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

We extend the min-max methods used in the critical point theory of differentiable functionals on smooth manifolds to the case of continuous functionals on a complete metric space. We study the topological properties of the min-max generated critical points in this new setting by adopting the methodology developed by Ghoussoub in the smooth case. Many old and new results are extended and unified and some applications are given.

In a coprime action, we study character correspondences between the invariant characters of the group and those of subgroups containing the fixed points subgroup. Some character degree consequences are obtained.

Let be a semigroup of measure preserving transformations on a measure space (Ω, ℱ, μ). The main result of the paper is the proof of a.e. convergence for the moving averages where {FIn} is a superadditive process and {In} is a sequence of cubes in satisfying the "cone-condition". The identification of the limit is given. A moving local theorem is also proved.

Given a strongly continuous semigroup of isometries ∪ acting on a Hilbert space ℋ, we construct an E0-semigroup α∪, the free E0-semigroup over ∪, acting on the algebra of all bounded linear operators on full Fock space over ℋ. We show how the semigroup αU⊗V can be regarded as the free product of α∪ and αV. In the case where U is pure of multiplicity n, the semigroup au, called the Free flow of rank n, is shown to be completely spatial with Arveson index +∞. We conclude that each of the free flows is cocycle conjugate to the CAR/CCR flow of rank +∞.

In this article we introduce and develop a theory of tensor products of JW-algebras. Since JW-algebras are so close to W*-algebras, one can expect that the W*-algebra tensor product theory will be actively involved. It is shown that if Mand N are JW-algebras with centres Z1 and Z2 respectively, then Z1 ⊗ Z2 is not the centre of the JW-tensor product JW() (see below for notation) ofMand N, in general. Also, the type decomposition of JW() has been determined in terms of the type decomposition of the JW-algebras M and N which, essentially, rely on the relationship between the types of the JW-algebra and the types of its universal enveloping Von Neumann algebra.

The well-known Tschirnhausen transformation, , eliminates the second term of the polynomial xn + axn-l + …. By a mere repeated application of this transformation, one can decide whether a given element of k[[x,y]] is prime (irreducible) or not. Here K is an algebraically closed field of characteristic 0. A generalised version of Hensel's Lemma is developed for the proofs. The entire paper can be understood by undergraduate students.

Coercive closed forms on L2-spaces are studied whose associated L2-semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.

The distance from an arbitrary rank-one projection to the set of nilpotent operators, in the space of k × k matrices with the usual operator norm, is shown to be sec(π/(k:+2))/2. This gives improved bounds for the distance between the set of all non-zero projections and the set of nilpotents in the space of k × k matrices. Another result of note is that the shortest distance between the set of non-zero projections and the set of nilpotents in the space of k × k matrices is .

Let ℝ+ denote the non-negative half of the real line, and let λ denote Lebesgue measure on the Borel sets of ℝn. A function φ: ℝn → ℝ+ is called a weight function if ʃℝn φ dλ = 1. Let (X, ℱ, μ) be a non-atomic, finite measure space, let ƒ: X → ℝ+, and suppose { Tν}ν∊ℝn is an ergodic, aperiodic ℝn-flow on X. We consider the weighted ergodic averages where is a sequence of weight functions. Sufficient as well as necessary and sufficient conditions for the pointwise, almost-everywhere convergence of are developed for a particular class of weight functions φk. Specifically, let {τk: ℝn → ℝn} be a sequence of measurable, non-singular maps with measurable, non-singular inverses such that the Radon-Nikodym derivatives dλ oτk /dλ and dλ oτk-1 / dλ are L∞ (ℝn), and such that τk and τ-1 map bounded sets to bounded sets. We examine convergence for the sequence where θk is an a.e.-convergent sequence of weight functions which are dominated by a fixed L1(ℝn) function with bounded support.

We continue our research on extension of complex varieties across closed subsets. While efforts are being made to deal with varieties of any dimensions, the paper primarily concerns 1-dimensional case, and the exceptional set is thus assumed to be connected with finite length. As applications of the main result, several corollaries are obtained with interesting features.