Amenable Banach algebras are characterised as those having approximate diagonals and having virtual diagonals. is relies on the projective tensor product introduced above.
De nition 3.14 LetAbe a Banach algebra. De ne thediagonal operator∆A : A⊗ˆA→
Aby setting
∆A(a⊗b) :=ab for alla⊗b∈A⊗ˆA.
De nition 3.15 (Runde, 2002, p44) LetAbe a Banach algebra. e elementM ∈
(
A⊗ˆA)∗∗is avirtual diagonal forAif, for alla∈A,
a·M=M·a and a·∆∗∗AM=a.
A bounded net(mα)αinA⊗ˆAis anapproximate diagonal forAif, for alla∈A,
a·mα−mα·a→0 and a·∆Amα →a.
eorem 3.16 (Runde, 2002, p45) ese three conditions are equivalent for the Ba- nach algebraA:
(i) Ais amenable.
(ii) Ahas an approximate diagonal. (iii) Ahas a virtual diagonal.
3.4 The weak containment property
C*-algebras are characterised as algebras of operators, and are also coupled to repre- sentation theory of groups. So it is that amenability goes full circle: from groups to Banach algebras to C*-algebras to representations and back to groups.
De nition 3.17 For a locally-compact groupG, the representationπisweakly con- tained in the representationτif all positive de nite functions associated withπare uniform limits on the compact subsets ofGof sums of positive functions associated withτ(Pavel, 2007).
Alternatively: for representationsπ, τ : G→ U(H),πis weakly contained inτ
(denotedπ⪯τ) if for everyξ∈ H, niteF⊆G, andϵ > 0there existsη1, . . . , ηn ∈
Hsuch that ⟨π(g)ξ, ξ⟩− n ∑ i=1 ⟨τ(g)ηi, ηi⟩ < ϵ
for allg∈F(Peterson, 2011).
We say thatGhas theweak containment propertywhen each irreducible unitary representation is weakly contained in the le regular representation (Pavel, 2007). Brie y, the recall that the le regular representation,π2, is given by:
π2(g)ξ:= ∑ t∈G ξtegt =∑ t∈G ξg−1tet
for allξ=∑t∈Gξtet ∈ℓ2(G).
In representation theory of groups, the following result is well-known. eorem 3.18 e following are equivalent for a groupG.
(i) Gis amenable.
(ii) Ghas the weak containment property.
(iii) e trivial representation ofGis weakly contained in the le regular represen- tation.
(iv) e algebrasC∗(G)andC∗r(G)are∗-isomorphic.
All but the last condition have been adequately de ned above. What areC∗(G)and
generality of inverse semigroups. Brie y, however,C∗(G)is theC*-algebra ofG, and is more explicitly called the C*-enveloping algebra of ℓ1(G). C∗(G) is de ned as the completion of ℓ1(G)under the supremum norm over all representations (ofG in some implied Hilbert spaceH). C∗r(G)is thereduced C*-algebra ofG, and is the norm closure ofπ2(ℓ1(G)).
Chapter 4
Amenability and Semigroups
Amenability is easily generalisable to semigroups, but results vary depending upon the choice of generalisation. Various nice theorems for groups, relating amenability to other properties, do not hold for semigroups.
4.1 Semigroups with finitely-additive measures
Recall De nition 2.1. At no point in the de nition was anything particularly “groupish” involved—no inverses, no identity element (or even associativity for that matter). e de nitioncouldapply immediately to any semigroup, using the natural le and right actions (le and right regular representations) of a semigroup on itself.
De nition 4.1 A semigroup S is le measurable if there exists a nitely-additive measure µ : P(S) → [0, 1] that is le invariant and has total measure 1 (Klawe, 1977; Paterson, 1988). Compare with De nition 2.1.
A le measurable semigroupS has the following properties, attributed to J. R. Sorenson (Klawe, 1977, p103).
(i) A homomorphic image ofSis not necessarily le measurable.
(ii) A le ideal ofSis not necessarily le measurable, but a right ideal must be le measurable.
Furthermore,
(i) A nite direct product of le measurable semigroups is le measurable. 50
(ii) A directed union of le measurable semigroups is le measurable. Compare these with eorem 2.30.
To motivate what will come later, the following result demonstrates why measur- ability is not a de nition of “semigroup amenability” in common use.
eorem 4.2 A non-trivial semigroup with zeroScannotbe le or right measurable. Proof is proof is similar to the one given by van Douwen (1992). For the semi- groupSwith zero0and nitely-additive measureµ,
1=µ(S) ∵total measure 1 =µ(0S) ∵le-invariance =µ({0}) =µ(0(S\ {0})) =µ(S\ {0}) ∵le-invariance =µ(S) −µ({0}) ∵ nitely additive =1−1=0,
a contradiction. is also holds for right-invariance. □
Here are some reasons that, beyond mere tradition, le measurability is not oen considered on semigroups:
• Zero elements are a common feature of many semigroups, and is utterly unsur- prising that the behaviour of a zero element is necessarily “volume-destroying”. erefore, we would expect that whatever de nition is used should account for that behaviour, and eorem 4.2 demonstrates that De ntion 2.1 is de cient in this regard.
• As yet, there appears to be no satisfying weakening of the three conditions. • Finitely-additive measure theory is less sophisticated and powerful than the
de nition that uses means. Given the power of the more sophisticated alter- native, it is not surprising that the de nition from Day (1957) is preferred. Le measurable semigroups will return later on.