Representations of dynamical systems on Banach spaces not containing \(l_{1}\) (Q2844844)

\textit{H. P. Rosenthal}'s dichotomy [Proc. Natl. Acad. Sci. USA 71, 2411--2413 (1974; Zbl 0297.46013)] shows that a Banach space does not contain an isomorphic copy of \(\ell^1\) if and only if every bounded sequence has a weak-Cauchy subsequence, and such a space is here called a Rosenthal space. The subtle interaction between analytic and topological properties that lies behind Rosenthal's dichotomy is here explored in the setting of topological dynamics and in particular the properties of spaces a given system can be modelled on. A topological group \(G\) acting continuously on a topological space \(X\) gives rise to a natural action of \(G\) on the Banach space \(C(X)\) -- an example of a Banach representation of the topological dynamical system \((X,G)\). Formally, such a pair \((X,G)\) has a continuous faithful representation on a Banach space \(V\) if there is a continuous co-homomorphism \(h\) from \(G\) into the group of self-isometries of a Banach space \(V\) and a weak*-continuous bounded \(G\)-equivariant map \(\alpha:X\to V^*\) with respect to the dual action \(G\times V^*\to V^*\) that is a topological embedding. The main result in this paper characterizes the `tame' dynamical systems as being those that have a faithful representation on a Rosenthal space. The tame systems are those with the property that every element of the enveloping semigroup is a Baire class one function. This work opens up the interesting idea of building Banach spaces with prescribed characteristics using topological dynamical systems.