# More about characterizations of exactness

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

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.