This volume sets forth a systematic development of continuous bounded cohomology with coefficients in Banach modules. It also gives a number of applications. Chapter one gives an overview of Banach modules and coefficient modules. This includes a discussion of spaces of \(L^\infty\) type as well as remarks on integration. Chapter two studies the notion of relative injectivity. Initially this is carried out in the context of spaces of continuous functions over proper \(G\)-spaces. Then it is set in the context of coefficient modules and related to Zimmer’s amenable actions. Chapter three begins the study of continuous bounded cohomology in earnest. The author begins very concretely and then moves on to the functorial approach. Various examples are given as well as the usual functorial properties. Additionally, the relation between continuous bounded cohomology and ordinary continuous cohomology is established via the comparison map. Chapter four is a smattering of techniques that are essential for later applications. Included are discussions of induction, \(L^p\) induction, and ergodicity leading to refined Lyndon-Hochschild-Serre exact sequences. Finally, chapter five presents a number of applications of the theory. For instance, it is related to rough actions, property (TT), quasi-morphisms, actions on the circle, and theorems on irreducible lattices.

Reviewer: Mark R.Sepanski (Waco)