We show that if$E$is a complex Banach space which contains no subspace isomorphic to$l$_{1}, then each infinite dimensional subspace of$E′$contains a normalized sequence which converges to zero for the weak star topology.

We find the exact value of the best possible constant associated with a covering problem on the real line.

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We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of ϖΓ and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.

We consider the operator,$f$(Δ) for Δ the Laplacian, on spaces of measures on the sphere in$R$^{$d$}, show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for the$L$^{2}-norm of$f$(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure on$R$^{$d$}with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.