The Banach Tarski Paradox and Amenability Lecture 19: Reiter s Property and the Følner Condition. 6 October 2011

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1 The Banach Tarski Paradox and Amenability Lecture 19: Reiter s Property and the Følner Condition 6 October 211

2 Invariant means and amenability Definition Let G be a locally compact group. An invariant mean is a linear functional m : L (G) R such that: 1. m(f ) if f 2. m(χ G ) = 1 3. m(g f ) = m(f ) for all g G and f L (G) Definition A locally compact group G is amenable if it admits an invariant mean.

3 Equivalence of amenability and the Følner Condition Last lecture we proved: G satisfies the Følner Condition = G admits an invariant mean Today we will define a property of locally compact groups called the Reiter Property. We ll then show (over 2 lectures): G admits an invariant mean = G satisfies the Reiter Property = G satisfies the Følner Condition and so obtain: Theorem (Følner, Greenleaf) Let G be a locally compact group. Then G admits an invariant mean if and only if G satisfies the Følner Condition.

4 Fubini s Theorem Fubini s Theorem provides conditions under which the order of integration may be reversed. A complete measure space is one in which all subsets of sets of measure zero are measurable (and have measure zero). The Lebesgue measure and counting measure are complete. Theorem Let (X, µ) and (Y, λ) be complete measure spaces. If f : X Y R is measurable (w.r.t. the product measure) then ( ) ( ) f (x, y) dλ dµ = f (x, y) dµ dλ A B B A for all measurable A X, B Y.

5 Fubini s Theorem A σ finite measure space is one which is the countable union of measurable sets of finite measure. Connected Lie groups are σ finite under Haar measure. Theorem Let (X, µ) and (Y, λ) be σ finite measure spaces. If either ( ) f (x, y) dλ dµ < or then A ( B A B B ( ) f (x, y) dµ dλ < A ) ( ) f (x, y) dλ dµ = f (x, y) dµ dλ B A

6 The spaces L 1 (G) and L (G) Let G be a locally compact group with Haar measure µ. Recall { } L 1 (G) := f : G R f is measurable and f dµ < L 1 (G) := L 1 (G)/{measurable functions f = µ a.e.} Then L 1 (G) is a normed linear space with norm (by abuse of notation) f 1 = f dµ L 1 (G) is the space of (equivalence classes of) integrable functions on G. Recall the dual space L (G) := { bounded linear functionals on L (G) } This is also a normed linear space, with the operator norm. G G

7 The spaces L 1 (G) and L (G) Lemma If f L 1 (G) and ϕ L (G) then f ϕ L 1 (G). Corollary There is an injective isometry L 1 (G) L (G), so we may regard L 1 (G) as a subspace of L (G). Proof. Let f L 1 (G). Define λ f : L (G) R by λ f (ϕ) = f ϕ dµ Then λ f is a bounded linear functional on L (G), λ f = f 1 and λ f = λ f if and only if f = f in L 1 (G). G

8 Means on L (G) Definition Let G be a locally compact group. An mean on L (G) is a linear functional m : L (G) R such that: 1. m(ϕ) if ϕ 2. m(χ G ) = 1 Lemma Let M be the set of all means on L (G). Then M is a subset of the unit ball in L (G). That is, a mean is a bounded linear functional of norm at most 1. Proof. Let ϕ L (G). Then ϕ χ G ϕ ϕ χ G Since m is linear and non-negative, it follows that m(ϕ) ϕ, as required.

9 The set L 1 (G) 1,+ We want a subset of L 1 (G) whose image in L (G) consists of means. Define L 1 (G) 1,+ = {f L 1 (G) : f 1 = 1 and f } Lemma Every f L 1 (G) 1,+ defines a mean m f : L (G) R via: m f (ϕ) := f ϕ dµ Lemma The image of L 1 (G) 1,+ is weak dense in M. Proof. Let m M. By definition of the weak topology on L (G), we need to show that there is a net {f n } in L 1 (G) 1,+ so that m fn (ϕ) m(ϕ) for all ϕ L (G). This apparently follows from the Hahn Banach Theorem. G

10 Reiter s Property The group G acts on L 1 (G) 1,+ via the left regular representation g f (x) = f (g 1 x) g G, f L 1 (G) 1,+ and x G Definition A locally compact group G satisfies Reiter s Property if for every every ε > and every compact subset K of G, there is an f L 1 (G) 1,+ such that for all k K k f f 1 ε Today we ll show that Reiter s Property implies the Følner Condition. Then in the next lecture(s) we ll prove: Theorem (Hulanicki 1965, Reiter 1966) Let G be a locally compact group. If G admits an invariant mean then G satisfies Reiter s Property.

11 Reiter s Property implies the Følner Condition Lemma Let G be a locally compact group with Haar measure µ. Let f, f L 1 (G) with f, f. For every t let and Then E t = {x G : f (x) t} E t = {x G : f (x) t} f f 1 = In particular, putting f = f 1 = µ(e t E t) dt µ(e t ) dt

12 Proof of Lemma Write χ t for the characteristic function of [t, ). Then for s, s [, ) so for all x G Hence by Fubini s Theorem χ t (s) χ t (s ) dt = s s χ t (f (x)) χ t (f (x)) dt = f (x) f (x) χ t f χ t f 1 dt = = G G = f f 1 χ t (f (x)) χ t (f (x)) dµ(x) dt f (x) f (x) dµ(x)

13 Proof of Lemma Now χ t f (x) = { 1 if f (x) t if f (x) < t Thus χ t f χ t f is the characteristic function of E t E t. So χ t f χ t f 1 = µ(e t E t) and thus as required. f f 1 = µ(e t E t) dt

14 Reiter s Property implies the Følner Condition Assume that the locally compact group G satisfies Reiter s Property. Let ε > and let Q be a compact subset of G containing e. Then K := Q 2 = {qq : q, q Q} is a compact subset of G. So by Reiter s Property, there exists f L 1 (G) 1,+ such that for all k K k f f 1 εµ(q) 2µ(K) Hence K k f f 1 dk εµ(q) 2

15 Reiter s Property implies the Følner Condition For t let E t be as in the Lemma. Then for each k K, ke t = {kx G : f (x) t} = {y G : f (k 1 y) t} = {y G : (k f )(y) t} Thus by the Lemma, for every k K Hence ( µ(e t ) K k f f 1 = µ(ke t E t ) µ(e t ) µ(ke t E t ) dt ) dk dt = k f f 1 dk εµ(q) K 2

16 Reiter s Property implies the Følner Condition We have ( ) µ(ke t E t ) µ(e t ) dk dt εµ(q) K µ(e t ) 2 By the Lemma, K µ(e t ) dt = f 1 = 1 so it follows that there is some t so that < µ(e t ) < and µ(ke t E t ) dk εµ(q) µ(e t ) 2 We will show that this set E t is a Følner set, for 2ε > and Q compact. Note that it s enough to consider compact sets containing e.

17 Reiter s Property implies the Følner Condition We have chosen t such that < µ(e t ) < and Define A = K µ(ke t E t ) µ(e t ) dk εµ(q) 2 { k K : µ(ke } t E t ) ε µ(e t ) We claim that Q AA 1. Assuming this, let a 1, a 2 A. Then (a 1 a2 1 E t E t ) ( (a 1 a2 1 E t a 1 E t ) (a 1 E t E t ) ) so as µ is G invariant µ(a 1 a2 1 t E t ) µ(a2 1 t E t ) + µ(a 1 E t E t ) = µ(a 2 E t E t ) + µ(a 1 E t E t ) Thus for all q Q 2εµ(E t ) µ(qe t E t ) 2εµ(E t )

18 Reiter s Property implies the Følner Condition We chose t such that < µ(e t ) < and and defined A = K µ(ke t E t ) µ(e t ) dk εµ(q) 2 { k K : µ(ke } t E t ) ε µ(e t ) Then µ(k \ A) < µ(q) 2. We use this prove the claim that Q AA 1. Let q Q. It is enough to show that µ(qa A) >. Now since K = Q 2 and Q contains e, from set inclusions we get µ(q) µ(qk K) µ(qa A) + µ(k \ A) + µ(q(k \ A)) The right-hand side equals µ(qa A) + 2µ(K \ A) < µ(qa A) + µ(q) thus µ(qa A) >, as required. This completes the proof.