Biholomorphic Mappings on Banach Spaces

We present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.

Fixed point properties for semigroups of nonexpansive mappings on convex sets in dual Banach spaces

It has been a long-standing problem posed by the first author in a conference in Marseille in 1990 to characterize semitopological semigroups which have common fixed point property when acting on a nonempty weak* compact convex subset of a dual Banach space as weak* continuous and norm nonexpansive mappings. Our investigation in the paper centers around this problem. Our main results rely on the well-known Ky Fan’s inequality for convex functions.

On metric characterizations of the Radon–Nikodým and related properties of Banach spaces

We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon–Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding.We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond.The paper also contains related characterizations of reflexivity and the infinite tree property.

A mean ergodic theorem of an amenable group action

We consider a sequence of weak Kadison–Schwarz maps τn on a von-Neumann algebra ℳ with a faithful normal state ϕ sub-invariant for each (τn, n ≥ 1) and use a duality argument to prove strong convergence of their pre-dual maps when their induced contractive maps (Tn, n ≥ 1) on the GNS space of (ℳ, ϕ) are strongly convergent. The result is applied to deduce improvements of some known ergodic theorems and Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space ℳ*.

Completions of Boolean algebras of projections and weak-star closures of C*-algebras on dual Banach spaces

Palmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

The Choquet–Deny Equation in a Banach Space

Let G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E. Given a probability measure μ on G, we study the Choquet–Deny equation π(μ)x = x, x ∈ E. We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law μ. The relation between the space of solutions of the Choquet–Deny equation in E and the space of bounded harmonic functions can be understood in terms of a construction resembling theW*-crossed product and coinciding precisely with the crossed product in the special case of the Choquet–Deny equation in the space E = B(L2(G)) of bounded linear operators on L2(G). Other general properties of the Choquet–Deny equation in a Banach space are also discussed.

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.