On Hilbert dynamical systems

Returning to a classical question in harmonic analysis, we strengthen an old result of Walter Rudin. We show that there exists a weakly almost periodic function on the group of integers ℤ which is not in the norm-closure of the algebra B(ℤ) of Fourier–Stieltjes transforms of measures on the dual group $\hat {\mathbb {Z}}=\mathbb {T}$, and which is recurrent. We also show that there is a Polish monothetic group which is reflexively but not Hilbert representable.

Infinite products of holomorphic mappings

LetXbe a complex Banach space,𝒩a norming set forX, andD⊂Xa bounded, closed, and convex domain such that its norm closureD¯is compact inσ(X,𝒩). Let∅≠C⊂Dlie strictly insideD. We study convergence properties of infinite products of those self-mappings ofCwhich can be extended to holomorphic self-mappings ofD. Endowing the space of sequences of such mappings with an appropriate metric, we show that the subset consisting of all the sequences with divergent infinite products isσ-porous.

Predual of the Multiplier Algebra of Ap(G) and Amenability

For a locally compact group G and 1 < p < ∞, let Ap(G) be the Herz-Figà-Talamanca algebra and let PMp(G) be its dual Banach space. For a Banach Ap(G)-module X of PMp(G), we prove that the multiplier space ℳ(Ap(G); X*) is the dual Banach space of QX, where QX is the norm closure of the linear span Ap(G)X of u f for u 2 Ap(G) and f ∈ X in the dual of ℳ(Ap(G); X*). If p = 2 and PFp(G) ⊆ X, then Ap(G)X is closed in X if and only if G is amenable. In particular, we prove that the multiplier algebra MAp(G) of Ap(G) is the dual of Q, where Q is the completion of L1(G) in the ‖ · ‖M-norm. Q is characterized by the following: f ∈ Q if an only if there are ui ∈ Ap(G) and fi ∈ PFp(G) (i = 1; 2, … ) with such that on MAp(G). It is also proved that if Ap(G) is dense in MAp(G) in the associated w*-topology, then the multiplier norm and ‖ · ‖Ap(G)-norm are equivalent on Ap(G) if and only if G is amenable.

The Dunford-Pettis Property for Symmetric Spaces

A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that ℓ1, c0 and ℓ∞ are the only symmetric sequence spaces with the Dunford- Pettis property, and that in the class of symmetric spaces on (0, α), 0 < α ≤ ∞, the only spaces with the Dunford-Pettis property are L1, L∞, L1 ∩ L∞, L1 + L∞, (L∞)◦ and (L1 + L∞)◦, where X◦ denotes the norm closure of L1 ∩ L∞ in X. It is also proved that all Banach dual spaces of L1 ∩ L∞ and L1 + L∞ have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces (L1 + L∞)◦ and (L∞)◦ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Köthe-Bochner spaces.

C*-Convexity and the Numerical Range

If A is a prime C*-algebra, a ∈ A and λ is in the numerical range W(a) of a, then for each ε > 0 there exists an element h ∈ A such that . If λ is an extreme point of W(a), the same conclusion holds without the assumption that A is prime. Given any element a in a von Neumann algebra (or in a general C*-algebra) A, all normal elements in the weak* closure (the norm closure, respectively) of the C*-convex hull of a are characterized.