The Banach Tarski Paradox and Amenability Lecture 20: Invariant Mean implies Reiter s Property 11 October 2012
Invariant means and amenability Definition Let be a locally compact group. An invariant mean is a linear functional m : L () R such that: 1. m(f ) 0 if f 0 2. m(χ ) = 1 3. m(g f ) = m(f ) for all g and f L () Definition A locally compact group is amenable if it admits an invariant mean.
Invariant mean implies Følner Today we will finish proving: Theorem (Følner, reenleaf) Let be a locally compact group. Then admits an invariant mean if and only if satisfies the Følner Condition. In Lecture 18 we showed: satisfies the Følner Condition = admits an invariant mean and in Lecture 19 we defined a property of locally compact groups called the Reiter Property and showed: satisfies the Reiter Property = satisfies the Følner Condition So today we will show (most of) admits an invariant mean = satisfies the Reiter Property
The spaces L p () and L () Let be a locally compact group with Haar measure µ. Let 1 p <. Recall L p () := { f : R f is measurable and } f p dµ < L p () := L p ()/{measurable functions f = 0 µ a.e.} Then L p () is a normed linear space with norm (by abuse of notation) ( ) 1/p f p = f p dµ In particular, L 1 () is the space of (equivalence classes of) integrable functions on. Recall the dual space L () := { bounded linear functionals on L () } This is also a normed linear space, with the operator norm.
Continuity of the left regular representation For 1 p the group acts on L p () via g f (x) = f (g 1 x) g, f L p () and x It is an exercise to show: Lemma Let f L p (). The mapping g g f from L p () is continuous if 1 p <. This mapping is not necessarily continuous for p =. Definition Let UCB() be the subspace of L () given by UCB() = {f L () g g f is continuous} The action on L () preserves UCB(). If is compact or discrete, then L () = UCB().
Means on L () and UCB() Definition Let be a locally compact group. An mean on L () is a linear functional m : L () R such that: 1. m(ϕ) 0 if ϕ 0 2. m(χ ) = 1 Similarly we can define a mean on UCB().
Means on L () and L 1 () We proved in Lecture 19: Lemma Let M be the set of all means on L (). Then M is a subset of the unit ball in L (). That is, a mean is a bounded linear functional of norm at most 1. Lemma If f L 1 () and ϕ L () then f ϕ L 1 (). Corollary There is an isometric embedding L 1 () L (), so we may regard L 1 () as a subspace of L (). The embedding is f λ f where for ϕ L () λ f (ϕ) = f ϕ dµ
The set L 1 () 1,+ In Lecture 19 we also defined a subset of L 1 () whose image in L () consists of means by L 1 () 1,+ = {f L 1 () : f 1 = 1 and f 0} Lemma Every f L 1 () 1,+ defines a mean m f : L () R via: m f (ϕ) := f ϕ dµ
Density of L 1 () 1,+ in M Lemma The image of L 1 () 1,+ is weak dense in M. Let m M L (). By definition of the weak topology on L (), we need to show that there is a sequence {f i } i in L 1 () 1,+ so that for all ϕ L () m fi (ϕ) m(ϕ) Use: Theorem (Hahn Banach) Let M be a linear subspace of a Banach space X and let x 0 X. Then x 0 is in M (the closure of M) if and only if there does not exist f X such that f (x) = 0 for all x M but f (x 0 ) 0.
Reiter s Property The group acts on L 1 () 1,+ via g f (x) = f (g 1 x) g, f L 1 () 1,+ and x Definition A locally compact group satisfies Reiter s Property if for every every ε > 0 and every compact subset K of, there is an f L 1 () 1,+ such that for all k K k f f 1 ε Today we ll prove: Theorem (Hulanicki 1965, Reiter 1966) Let be a locally compact group. If admits an invariant mean then satisfies Reiter s Property. using the proof in Bekka de la Harpe Valette.
Properties of the set L 1 () 1,+ Lemma L 1 () 1,+ is a convex subset of L 1 (). Proof. If f, g L 1 () 1,+ and t [0, 1] then tf + (1 t)g 0 and tf + (1 t)g 1 = tf + (1 t)g dµ = tf + (1 t)g dµ = t f dµ + (1 t) g dµ = t f 1 + (1 t) g 1 = 1 so tf + (1 t)g L 1 () 1,+ as required for convexity.
Properties of the set L 1 () 1,+ Lemma L 1 () 1,+ is closed under convolution, where (f g)(x) := f (y)g(y 1 x) dµ(y) for all f, g L 1 () 1,+ and x. Proof. We have f g 0. By Fubini s Theorem, since f and g are integrable ( ) ( ) f g dµ = f dµ g dµ hence f g 1 = f 1 g 1 = 1
Properties of the set L 1 () 1,+ Definition Let UCB() be the subspace of L () given by UCB() = {f L () g g f is continuous} Lemma If f L 1 () 1,+ and ϕ L () then f ϕ = f (y)(y ϕ) dµ(y) is in UCB(). Proof. Exercise.
Invariant mean implies Reiter Property Suppose admits an invariant mean m on L (). Then m restricts to an invariant mean on UCB(). Lemma For all f L 1 () 1,+ and all ϕ UCB(), m(f ϕ) = m(ϕ) Assuming the lemma, let {f i } i be a sequence in L 1 () 1,+ such that supp(f i ) {e}. Then for all ϕ L () and f L 1 () 1,+ lim i f f i ϕ f ϕ = 0 so since means are continuous and by the result of the lemma m(f ϕ) = lim i m(f f i ϕ) = lim i m(f i ϕ) Thus for all ϕ L () and all f, f L 1 () 1,+ m(f ϕ) = m(f ϕ) (1)
Invariant mean implies Reiter Property We have m : UCB() R an invariant mean. Fix a function f 0 L 1 () 1,+. Define a mean m on L () by m(ϕ) = m(f 0 ϕ) Then m(ϕ) 0 if ϕ 0 and m(χ ) = m(f 0 χ ) = m(χ ) = 1 by the lemma. For all f L 1 () 1,+ and ϕ L () we have by (1) m(f ϕ) = m(f 0 f ϕ) = m(f 0 ϕ) = m(ϕ). That is, m is a topological invariant mean on L (), meaning that m is a mean so that for all f L 1 () 1,+ and ϕ L () m(f ϕ) = m(ϕ).
Invariant mean implies Reiter Property Since L 1 () 1,+ is weak dense in the set of all means on L (), there exists a sequence {f i } in L 1 () 1,+ converging to m in the weak topology. That is, for all ϕ L (), m fi (ϕ) m(ϕ) Let f L 1 () 1,+. From the definition of convolution we get for all ϕ L (), m f fi (ϕ) = m fi (f ϕ) thus as m is a topological invariant mean m (f fi f )(ϕ) = m f fi (ϕ) m fi (ϕ) m(f ϕ) m(ϕ) = 0 In other words, (f f i f i ) converges to 0 in the weak topology on L 1 (), where we consider L 1 () as a subspace of L ().
Invariant mean implies Reiter Property Consider E = f L 1 () 1,+ L 1 () Since the set L 1 () 1,+ is convex, the set E is locally convex. Define Σ = {(f g g) f L 1 () 1,+ : g L 1 () 1,+ } E The set Σ is convex (check), and since (f f i f i ) 0 in the weak topology on each L 1 (), the closure of Σ in the weak topology on E (which is just the product of the weak topologies on each L 1 ()) contains 0. Since E is locally convex and Σ is convex, the closure of Σ in the weak topology is the same as the closure of Σ in the product of the norm topologies. So there exists a sequence {g j } j in L 1 () 1,+ such that for all f L 1 () 1,+ lim j f g j g j 1 = 0
Invariant mean implies Reiter Property There exists a sequence {g j } j in L 1 () 1,+ such that for all f L 1 () 1,+ lim j f g j g j 1 = 0 (2) Since each g j has bounded norm, this holds uniformly for f in compact (w.r.t. norm) subsets of L 1 (). Let Q be a compact neighbourhood of e. Let ε > 0. Fix f L 1 () 1,+. Since the map g g f is continuous from to L 1 (), the set {q f q Q} is compact in L 1 (). So by Equation (2) holding uniformly on compact subsets, there exists j such that for all q Q (q f ) g j g j 1 < ε Put g = f g j. Note that q g = (q f ) g j. Then for all q Q (q g) g 1 (q f ) g j g j 1 + (f g j ) g j 1 < 2ε We have established the Reiter Property.