The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $\left(L{B}^{\infty }\right)$ for the dual space of the space of germs of holomorphic functions on that compact set.

Let A be a commutative Banach algebra and let ${\Sigma }_A$ be its structure space. The norm spectrum σ(f) of the functional f ∈ A* is defined by $\sigma \left(f\right)=\overline{f·a:a\in A}\cap {\Sigma }_A$, where f·a is the functional on A defined by ⟨f·a,b⟩ = ⟨f,ab⟩, b ∈ A. We investigate basic properties of the norm spectrum in certain classes of commutative Banach algebras and present some applications.

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C*-algebra theory.

We study the numerical radius of Lipschitz operators on Banach spaces. We give its basic properties. Our main result is a characterization of finite-dimensional real Banach spaces with Lipschitz numerical index 1. We also explicitly compute the Lipschitz numerical index of some classical Banach spaces.

It is shown that a sequentially complete topological vector space X with a compact Schauder basis has WSPAP (see Definition 2) if and only if X has a pseudo-homogeneous norm bounded on every compact subset of X.

We obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.

We prove a number of fundamental facts about the canonical order on projections in C*-algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of their lower bounds) decreasing sequences. Under the further assumption that the order is countably downwards closed, we show how to characterize greatest lower bounds of finite collections of projections, and their existence, using the norm and spectrum of simple...

Let $G$ be a locally compact group, and let $\parallel {\parallel }_{\infty }^B$ be a function norm on $L^1(G{)}_{\mathrm{loc}}$ such that the space $L^{\infty }(G,B)$ of all locally integrable functions with finite $\parallel {\parallel }_{\infty }^B$-norm is an invariant solid Banach function space. Consider the space $L_{\mathrm{RUC}}(G,B)$ of all functions in $L^{\infty }(G,B)$ of which the right translation is a continuous map from $G$ into $L^{\infty }(G,B)$. Characterizations of the case where $L_{\mathrm{RUC}}(G,B)$ is a Riesz ideal of $L^{\infty }(G,B)$ are given in terms of the order-continuity of $\parallel {\parallel }_{\infty }^B$ on certain subspaces of $L^{\infty }\left(G\right)$. Throughout the paper, the discussion is carried out in the context...

The order topology ${\tau }_o\left(P\right)$ (resp. the sequential order topology ${\tau }_{os}\left(P\right)$) on a poset P is the topology that has as its closed sets those that contain the order limits of all their order convergent nets (resp. sequences). For a von Neumann algebra M we consider the following three posets: the self-adjoint part $M_{sa}$, the self-adjoint part of the unit ball $M{¹}_{sa}$, and the projection lattice P(M). We study the order topology (and the corresponding sequential variant) on these posets, compare the order topology to the other...