Table Of ContentM. Roelands
Amenability in Positivity
Master’s Thesis, defended on August 14, 2012
Thesis Advisor: Dr. M. F. E. de Jeu
Mathematical Institute, Leiden University
Dedicated to my grandfather W. H. Roelands,
in loving memory.
Contents
1 Introduction and overview 1
2 Characterizations of amenable groups 4
2.1 Integration theory on locally compact groups . . . . . . . . . . . . . . . . . 4
2.1.1 Integrable functions on G . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 L1(G) as an ideal of M(G) . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 The dual space of L1(G) . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.4 Amenable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Modules over a Banach algebra . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Cohen’s factorization theorem . . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Neo-unital modules and extensions of derivations to larger modules 27
2.3 Johnson’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Johnson’s theorem in an ordered context . . . . . . . . . . . . . . . 37
3 Hochschild cohomology groups of Banach algebras 38
3.1 Constructing Hochschild cohomology groups of order n ∈ N . . . . . . . . 38
+
3.1.1 Tensor products of Banach spaces . . . . . . . . . . . . . . . . . . . 44
3.1.2 The amenability of A in terms of Hn(A,E∗) . . . . . . . . . . . . . 49
3.2 ThetrivialityofHn(A,E∗)fororderedBanachalgebrasandregularBanach
A-bimodules E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Hochschild cohomology groups for Banach lattice algebras . . . . . . . . . 55
3.3.1 Tensor products of Banach lattices . . . . . . . . . . . . . . . . . . 59
3.4 The triviality of Hn(A,E∗) for Banach lattice algebras and regular Banach
r
lattice bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4 Concluding remarks 84
5 Acknowledgements 86
6 References 87
1 Introduction and overview
This thesis is about amenability. There are several contexts in which this property can
be used and we will discuss what these are and how they relate to one another and,
simultaneously, give an overview of what is done in this thesis. The notion of amenable
groups arose in the first half of the 20th century in the context of the famous Banach-
Tarski paradox:
Theorem 1.1 (Banach-Tarski) Every closed ball in R3 is paradoxical.
In words, this means that every closed ball in R3 can be divided into a finite amount
of disjoint subsets which can be put together again and yield two identical copies of the
original ball. Reassembling these disjoint subsets is done by letting the special orthogonal
group SO(3) of R3 act on them and the key idea in the proof of this paradox lies in the
observation that this group has a subgroup which is isomorphic to the free group F with
2
two generators.
Definition: Let G be a group. Then G is paradoxical, or G allows a paradoxical
decomposition if there are pairwise disjoint subsets A ,...,A ,B ,...,B in G along
1 n 1 m
with g ,...,g ,h ,...,h ∈ G such that
1 n 1 m
n m
G = g ·A and G = h ·B .
k k k k
k=1 k=1
[ [
This free group is paradoxical when acting on itself, for if a and b are the generators of
F and we define
2
W(y) := {x ∈ F : x starts with y} (y = a,a−1,b,b−1),
2
then we can write F as the disjoint union
2
F = {e }∪W(a)∪W(b)∪W(a−1)∪W(b−1)
2 F2
and if x ∈ F \W(a), then a−1x ∈ W(a−1), so x ∈ aW(a−1); hence F = W(a)∪aW(a−1).
2 2
Similarly, one finds that F = W(b)∪bW(b−1). Now the class of groups that do not allow
2
these paradoxical decompositions were characterized in order to omit the behavior stated
in the Banach-Tarski paradox:
Theorem 1.2 A discrete group G is not paradoxical if and only if there exists a functional
m ∈ ℓ∞(G)∗ that satisfies
i) m(1) = kmk = 1;
ii) m(δ ∗φ) = m(φ) (g ∈ G,φ ∈ ℓ∞(G)).
g
Here the function δ ∗φ is just the translation of φ over g and is defined by
g
δ ∗φ(h) := φ(g−1h)
g
and the notation comes from a convolution of the discrete measures on ℓ∞(G). This
characterization can be found in [15, Cor. 0.2.11]. The functional m ∈ ℓ∞(G) is called a
left invariant mean on G and a discrete group G is said to be amenable, apparently
1
as a pun, if such a mean exists on G. As an example, all finite groups are amenable, for
the left invariant in question mean would be defined by
1
m(f) := f(g).
|G|
g∈G
X
Also, all compact groups are amenable as we can define the left invariant mean again in
this case:
m(f) := f(g)dm (g).
G
ZG
Finally, all abelian groups are amenable as well. This is stated in [15, Ex. 1.15].
From a functional analytic point of view, we would like to extend the notion of
amenability to general locally compact groups and this is done in section 2.1.4. A theorem
of Johnson’s characterizes the amenability of locally compact groups G in terms of the
triviality of a specific cohomology group on the Banach space of integrable functions on
G.
In a more general setting, a cochain complex is a sequence (A ,d ) of modules
n n n≥0
A together with homomorphisms
n
d : A → A
n n n+1
such that d ◦d = 0 for all n ≥ 1 and is denoted by
n n−1
{0} → A →d0 A →d1 ··· → A →dn A d→n+1 ···
0 1 n n+1
Since we have that Im(d ) ⊂ ker(d ), we can consider the quotient modules
n−1 n
H(A ) := ker(d )/Im(d ) (n ≥ 1)
n n n−1
which are referred to as the cohomology groups of A . These types of complexes can
n
be constructed for Banach spaces as well, as is done in section 3.1.
Under pointwise operations, the real valued integrable functions on G allow a vector
space ordering and the first goal of this thesis will be to construct a characterization
of amenable locally compact groups, similar to Johnson’s theorem, purely in an ordered
context.
Johnson’s theorem also induces the notion of an amenable Banach algebra. A remark-
able result, proven in section 3.1.2, is that a Banach algebra which is amenable yields
trivial cohomology groups for all n ≥ 1. The proof uses tensor products of Banach spa-
ces and section 3.1.1 is, partially as a general reminder, devoted to providing sufficient
background knowledge about them. The second goal of this thesis is to investigate under
which conditions an ordered Banach algebra yields trivial cohomology groups for all n ≥ 1
as well.
As was mentioned above, the vector space order onthe real valued integrable functions
on G, in addition, turns this space into a Banach lattice algebra and the third goal of
this thesis is to construct a cochain complex specifically for Banach lattice algebras such
that the corresponding cohomology groups propose an alternative notion of amenability
and have similar properties as in the case of general Banach algebras. In the first part of
section 3.3 this construction is described. As for the triviality of the cohomology groups
for all n ≥ 1 in this respect, along the lines of the previous findings concerning Banach
algebras, we shall consider tensor products of Banach lattices which will be studied, in
2
depth, in section 3.3.1. The main result here will be that under the assumption that
we have an alternatively amenable Banach lattice algebra, we obtain trivial cohomology
groups for all n ≥ 1 when using these tensor products of Banach lattices.
Just in order to clarify things, all vector spaces considered in section 2.1 until 3.2 are
regarded as complex and in the remainder, the vector spaces will be real.
”A mathematician is a device for turning coffee into theorems.”
- Paul Erd˝os -
3
2 Characterizations of amenable groups
The main goal in this chapter will be to extend the notion of amenable discrete groups
to general locally compact groups and to characterize them using Johnson’s theorem.
Finally, we will place this result in an ordered context.
2.1 Integration theory on locally compact groups
In this section we will study the theory of integrable functions defined on locally compact
groups, in terms of convolution products, dual spaces and ideals. We start with some
elementary properties of topological groups and will assume that all groups here are
Hausdorff.
Lemma 2.1 For a topological group G we have:
i) every open subgroup of G is clopen;
ii) if A and B are compact sets in G, then so is AB.
Proof: If H is an open subgroup of G, then so are all its cosets gH, because the map
x 7→ gx is a homeomorphism on G. If g ∈ G \ H, then for all h ∈ H we have that
gh ∈ G\H and since e ∈ H, we conclude that G\H = gH; hence G\H is open.
g∈G\H
Equivalently, we find that H is closed. As for the second property, since AB is the image
S
of the compact set A×B under the continuous map (g,h) 7→ gh, we conclude that AB
must be compact too.
Lemma 2.2 Let G be a locally compact group. The there exists a subgroup H of G that
is clopen and σ-compact.
Proof: Since G is locally compact, there exists a compact neighborhood N of e. By the
continuity of the map g 7→ g−1, we find that N−1 is a compact neighborhood of e as well.
The intersection U := N ∩N−1 now yields a symmetric and compact neighborhood of e.
For each n ∈ N we define U := n U and let H := ∞ U . It is a straightforward
+ n k=1 n=1 n
verification to see that H is a group. If x ∈ H, then there is a number n ≥ 1 such that
Q S
x ∈ U and for an open set V ⊂ U which contains e, we have that x ∈ Vx ⊂ U ⊂ H.
n n+1
Now V is also open since g 7→ gx is a homeomorphism; hence H is open and Lemma 2.1
now implies that H is clopen. Moreover, each U is also compact by Lemma 2.1 and we
n
conclude that H is σ-compact.
For a locally compact group G, we define M(G) to be the space of finite and regular Borel
measures on G. By the regularity of µ we mean that its variation |µ|, which is defined by
n n
|µ|(E) := sup |µ(E )| : n ∈ N , E ∈ B(G), E = E (E ∈ B(G)),
k + k k
( )
k=1 k=1
X ]
is regular, where B(G) denotes the Borel sets of G. By the Riesz representation theorem,
the space M(G) can isometrically be identified with C (G)∗, which is a Banach space,
0
with respect to the norm
n n
kµk := sup |µ(E )| : n ∈ N , E ∈ B(G), G = E (µ ∈ M(G)).
k + k k
( )
k=1 k=1
X ]
4
