A geometric property is introduced, and it is proved that Banach spaces with this property must carry weakly amenable algebras of approximable operators. These results are also applied to particular examples of Banach spaces, including the James spaces and the Tsirelson space.

Guided by the Hopf fibration, a family (indexed by a positive constant $K$ ) of right invariant Riemannian metrics on the Lie group $S^3$ is singled out. Using the Yasuda–Shimada paper as an inspiration, a privileged right invariant Killing field of constant length is determined for each $K > 1$ . Each such Riemannian metric couples with the corresponding Killing field to produce a $y$ -global and explicit Randers metric on $S^3$ . Employing the machinery of spray curvature and Berwald's formula, it is proved directly that the said Randers metric has constant positive flag curvature $K$ , as predicted by Yasuda–Shimada. It is explained why this family of Finslerian space forms is not projectively flat.

The set of essentially Hankel operators is introduced and some of its properties are investigated. It is shown in particular that the set contains some operators not of the form ‘Hankel plus compact’, even when restricted to the class of essentially Hankel operators with trivial (Fredholm) index function.

The paper shows that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (that is, is tangent to a smooth variation through harmonic maps). This provides one of the few known answers to the problem of integrability, which was raised in different contexts of geometry and analysis. It implies that the Jacobi fields form the tangent bundle to each component of the manifold of harmonic maps from $S^2$ to ${\bb C} P^2$ thus giving the nullity of any such harmonic map; it also has a bearing on the behaviour of weakly harmonic $E$ -minimizing maps from a 3-manifold to ${\bb C} P^2$ near a singularity and the structure of the singular set of such maps from any manifold to ${\bb C} P^2$ .

In this paper we shall prove that the measure algebra $M(G)$ of a locally compact group $G$ is amenable as a Banach algebra if and only if $G$ is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that $M(G)$ is not amenable in the case where the group $G$ is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.

The paper studies the dual algebras of localization Roe algebras over proper metric spaces and develops a localization version of Paschke duality for $K$ -homology. It is shown that the localization $K$ -homology groups are isomorphic to Kasparov's $K$ -homology groups for the Rips complex of proper metric spaces with bounded geometry. It follows that the obstruction groups to the coarse Baum–Connes conjecture can also be derived from the dual localization algebras.

Parthasarathy and Sunder have proved that the set of coherent vectors associated with the indicator functions of Borel sets is total in the boson Fock space $\Gamma(L^2({\bb R}^+;{\bb C}))$ . The paper studies the space generated by coherent vectors associated with the union of $n$ intervals. A complete characterization is given of their orthogonal space in terms of their chaos expansion. Parthasarathy and Sunder's result is recovered in a very simple way. In the cases of the Brownian motion or Poisson process interpretation of the Fock space, the result characterizes those random variables that are orthogonal to the exponential of any sum of $n$ increments of the Brownian motion or Poisson process.

In this paper, we are mainly concerned with $n$ -dimensional simplices in hyperbolic space ${\bb H}^n$ . We will also consider simplices with ideal vertices, and we suggest that the reader keeps the Poincaré unit ball model of hyperbolic space in mind, in which the sphere at infinity ${\bb H}^n(\infty)$ corresponds to the bounding sphere of radius 1. It is known that all hyperbolic simplices (even the ideal ones) have finite volume. However, explicit calculation of their volume is generally a very difficult problem (see, for example, [1] or [16]). Our first theorem states that, amongst all simplices in a closed geodesic ball, the simplex of maximal volume is regular. We call a simplex regular if every permutation of its vertices can be realized by an isometry of ${\bb H}^n$ . A corresponding result for simplices in the sphere has been proved by Böröczky [4].

Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems are proved for nonexpansive mappings, including an improvement of a result of Lions. A modification of Rockafellar’s proximal point algorithm is obtained and proved to be always strongly convergent. The ideas of these algorithms are applied to solve a quadratic minimization problem.

Let $D$ be an open set in euclidean space ${\bb R}^m$ with non-empty boundary $\partial D$ , and let $p_D : D \times D \times; [0,\infty)) \longrightarrow {\bb R}$ be the Dirichlet heat kernel for the parabolic operator ${-}\Delta + \partial/\partial t$ , where ${-}\Delta$ is the Dirichlet laplacian on $L^2(D)$ . Since the Dirichlet heat kernel is non-negative, we may define the (open) set function \renewcommand{\theequation}{1.1} \begin{equation} P_D = \int\nolimits^{\infty}_0 \int\nolimits_D \int\nolimits_D p_D (x,y;t)\,dx\,dy\,dt. \end{equation} We say that $D$ has finite torsional rigidity if $P_D < \infty$ . It is well known that if $D$ has finite volume, then $D$ has finite torsional rigidity [11]. As we shall see, the converse is not true. The main purpose of this paper is to obtain necessary and sufficient conditions on the geometry of $D$ to guarantee finite torsional rigidity and to gain some understanding of the behaviour of the expected lifetime of brownian motion in a certain natural class of domains that do not have finite torsional rigidity.