In this paper, we show first that any prime (or semiprime) element of a commutative Banach algebra must have closed range. As a corollary, we find that in a commutative radical Banach algebra, all primes are zero divisors; indeed, all semiprimes are zero divisors (see below for the definition of semiprimeness). Our result is also true of a semiprime that is in the centre of a noncommutative Banach algebra.

The proof is fairly simple and entertaining, and we obtain a result that is helpful for the ambitious classification of elements in commutative radical Banach algebras being attempted by Marc Thomas. It is also related to the unbounded Kleinecke–Shirov conjecture.

Necessary and sufficient conditions for Lefschetz fibrations over D2 and S2 to carry spin structures are presented. The conditions are given in terms of vanishing cycles and homology sections.

We present a simple example of an integrable function for which the integral is not differentiable almost everywhere in the strong sense. A first example of such a function was given by S. Saks in 1935. Our construction is considerably more simple, due to the use of a remarkable theorem of P. X. Gallagher.

Let X be a reflexive Banach space, and let C ⊂ X be a closed, convex and bounded set with empty interior. Then, for every δ > 0, there is a nonempty finite set F ⊂ X with an arbitrarily small diameter, such that C contains at most δ · |F| points of any translation of F. As a corollary, a separable Banach space X is reflexive if and only if every closed convex subset of X with empty interior is Haar null.

It is shown that a hypercentral group that has all subgroups subnormal and every non-nilpotent subgroup of bounded defect is nilpotent. As a consequence, a hypercentral group of length at most ω in which every subgroup is subnormal is nilpotent.

It is known that if a is an algebraic element of a Banach algebra A, then its spectrum σ(a) is finite, and there exists γ > 0 such that the Hausdorff distance to spectra of nearby elements satisfies

(formula here)

We prove that the converse is also true, provided that A is semisimple.

In a previous paper, the author proved a recent Wiener–Hopf formula, established by Alili and Doney, by means of a path transform. We derive here two other results using this path transform.

The nonexistence of a submersion from the 23-sphere to the Cayley projective plane is proved.

Consider the following problem: given complex numbers a1, …, an, find an L∞ function f of minimum norm whose Fourier coefficients ck(f) are equal to ak for k between 0 and n. We show the uniqueness of this function, and we estimate its norm. The operator-valued case is also discussed.

A mean-value theorem, an inverse mapping theorem and an implicit mapping theorem are established here in a class of complex locally convex spaces, including the spaces of test functions in distribution theory. Our main tool is the integral formula and the invariance of the domain, derived from topological degrees, rather than from fixed points of contractions in Banach spaces.

The weak compactness of the composition operator CΦ(f) = f ○ Φ acting on the uniform algebra of analytic uniformly continuous functions on the unit ball of a Banach space with the approximation property is characterized in terms of Φ. The relationship between weak compactness and compactness of these composition operators and general homomorphisms is also discussed.

We show that any gauge-equivalence class of solutions (∇, Φ) of the monopole equation F∇ = *d∇Φ, defined on an open set Ω of S3 (equipped with its standard metric), induces canonically a holomorphic function on the complex 2-dimensional space of all geodesics on S3 entirely contained in Ω.

Let G be a compact connected Lie group and K a maximal rank subgroup of G. The homogeneous space G/K has the S1-action defined by left translations induced from a homomorphism from S1 to G. In this paper, we study a problem on the realization of some deformation of the cohomology algebra H*(G/K; [ ]p) by the S1-equivariant cohomology of G/K. In consequence, for the case where G is a classical Lie group, it follows that there exists at most one essentially different homomorphism from S1 to G which realizes a given deformation, and that the homomorphism is controlled by an appropriate equation in one indeterminate.

Let F(z) = [sum ]∞n=0anzn be an analytic function in the unit disc satisfying

[formula here]

Then [formula here], which is familiar as Paley's inequality. In this paper, an analogue of this inequality with respect to the Jacobi expansions is established.

We study closed subsets in the plane which intersect each line in at least m points and at most n points, for which we try to minimize the difference n – m. It is known that m cannot be equal to n. The results in this paper show that for every even number n there exist closed sets in the plane for which m = n – 2.

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. The asymptotics of the number of geodesics in M starting from and returning to a given cusp, and of the number of horoballs at parabolic fixed points in the universal cover of M, are studied in this paper. The case of SL(2, ℤ), and of Bianchi groups, is developed.

Let X be a compact manifold with a smooth action of a compact connected Lie group G. Let L → X be a complex line bundle. Using the Cartan complex for equivariant cohomology, we give a new proof of a theorem of Hattori and Yoshida which says that the action of G lifts to L if and only if the first Chern class c1(L) of L can be lifted to an integral equivariant cohomology class in H2G(X; ℤ), and that the different lifts of the action are classified by the lifts of c1(L) to H2G(X; ℤ). As a corollary of our method of proof, we prove that, if the action is Hamiltonian and ∇ is a connection on L which is unitary for some metric on L, and which has a G-invariant curvature, then there is a lift of the action to a certain power Ld (where d is independent of L) which leaves fixed the induced metric on Ld and the connection ∇[otimes ]d. This generalises to symplectic geometry a well-known result in geometric invariant theory.

This paper presents explicit constructions of universal ℝ-trees as certain spaces of functions, and also proves that a 2ℵ0-universal ℝ-tree can be isometrically embedded at infinity into a complete simply connected manifold of negative curvature, or into a non-abelian free group. In contrast to asymptotic cone constructions, asymptotic spaces are built without using the axiom of choice.

The purpose of this note is to establish a uniform estimate for the mass function ℙ(Sm = y) of an integer-valued random walk when y → ∞ and (y − mμ)/√m → ∞, where μ is the mean of the step distribution. (The local central limit theorem provides such an estimate when (y − mμ)/√m is bounded.) The assumptions are that the mass function p of the step distribution is regularly varying at ∞ with index −κ, where κ > 3, and that [sum ]∞n=0nκ′p(−n) < ∞ for some κ′ > 2. From this result, a ratio limit theorem is derived, and this in turn is applied to yield some new information about the space–time Martin boundary of certain random walks.