More about characterizations of exactness

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.