Following up an earlier post with some notes on the three papers below:

- Invariant expectations and vanishing of bounded cohomology for exact groups by Douglas and Nowak [DN]
- Amenable actions, invariant means and bounded cohomology by Brodzki, Niblo, Nowak and Wright [BNNW]
- A note on topological amenability by Monod. [M]

All of these papers focus on the question of characterizing in “homological” terms what it is for a discrete group \( G \) to be exact (or, more generally, to act amenably on some compact space — it is known that exactness is equivalent to the amenability of the action of \( G \) on its Stone-Cech compactification \( \beta G \)).

A necessary step along the way (again, in all the papers) is to relate the notion of exactness to some kind of “invariant mean”. This follows a path explained by Johnson for ordinary amenability (Johnson, Barry Edward. 1972. Cohomology in Banach algebras. Providence, R.I.: American Mathematical Society.)

The paper [M] gives the greatest number of equivalent conditions (it is a legal requirement that all papers on amenability show that many conditions are equivalent). In particular let us consider the appropriate notion of “invariant mean”. This is an element \( \phi \) of the bidual \( A^{**} \), where \( A \) is the algebra of continuous functions on \( \beta G \) with values in \( \ell^1 G \) (also equivalent to the algebra of unconditionally convergent formal series \( \sum_g f_g [g] \), with \( f_g \in \ell^\infty(G) \)); \( \phi \) must be \( G \)-invariant and must sum to the constant function \( 1 \in \ell^\infty(G) \).

**Aside:** Monod also gives a couple of interesting alternative characterizations of \( A \):

- The space of compact operators on \(\ ell^1(G) \)
- The space of weak-\( * \) – to – weak continuous operators on \( \ell^\infty(G) \)

The characterizations in [BNNW] seem quite close to that of [M]. Their invariant means are elements of a double dual \( W_0^{**} \), where \( W_0 \) is defined a little differently but appears to be the subspace of \( A \) consisting of elements that sum to a multiple of 1.

In [DN] the double dual of \( W_0 \) is taken in a more ambitious sense, as \( \hom(\hom(W_0,C),C) \), where \( C \) is the Banach space \( \ell^\infty(G) \). But then one looks inside this double dual at the weak-\( * \) closure, in an appropriate sense, of the members of \( W_0 \) itself. Now, let \( R \) be some suitable ring of endomorphisms of the Banach space \( C \) (e.g., the translation algebra). Then both \( C \) and \( hom(W_0,C) \) are \( R \)-modules and the elements of \( \hom(\hom(W_0,C),C) \) coming from \( W_0 \) are \( R \)-module maps. Thus, it seems to me, one might as well restrict to the subspace of \( R \)-module maps from the start, and then much of the extra “size” of the double dual goes away. I think this may make a connection between the [DN] approach and the other two.