On operator amenability ofFourier-Stieltjes algebras

Nico Spronk, U. of Waterloo

Banach algebras 2019U. of Manitoba

July 17, 2019

Fourier algebra: non-commutative Pontryagin duality

G – locally compact group with left Haar measure

HvN alg. L∞(G ) C0(G )? _oo C∗r (G ) �� // VN(G )

predual L1(G )

λ

55kkkkkkkkkkkkkkkkkkA(G )

λ

SSSSSSSSSSSSSSSS

iiSSS

group “dual group”

λ : G → U(L2(G )), λ(s)h(t) = h(s−1t)integrated form: λ : L1(G )→ B(L2(G )), λ(f )h =

∫G f (s)λ(s)h ds

VN(G ) = λ(G )′′, coproduct extends λ(s) 7→ λ(s)⊗ λ(s)A(G ) = {s 7→ 〈λ(s)h, g〉 : h, g ∈ L2(G )} – Fourier algebra

λ : A(G )→ C0(G ) – natural injection, algebra homomorphism

Fourier-Stieltjes algebra: duality with universal rep’n

dual M(G )

$

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TTTTTTTT

TTTB(G )

$

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j

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HC* alg. C0(G ) �� // Cb(G ) MC∗(G ) C∗(G )? _oo

$ =⊕

π cyclic π : G → U(H$) – universal representation

integrated form: $ : M(G )→ B(H$), $(µ)ξ =∫

G $(s)ξ dµ(s)

C∗(G ) = $(L1(G ))‖·‖ ∈ B(H$).

coproduct extends:∫

G f (s)$(s) ds 7→∫

G f (s)$(s)⊗$(s) ds :C∗(G )→ M(C∗(G )⊗min C∗(G ))

B(G ) = {s 7→ 〈π(s)ξ, η〉 : π cts. unitary rep’n, ξ, η ∈ Hπ}= {s 7→ 〈$(s)ξ, η〉 : ξ, η ∈ H$} – Fourier-Stieltjes algebra

$ : B(G )→ Cb(G ) – natural injection, algebra homomorphism

Summary

A(G ) = “L1(G )”, B(G ) = “M(G )”

Ideals:L1(G ) ∼= {ν ∈ M(G ) : ν << Haar}CM(G ), by left invariance;A(G )C B(G ), by Fell’s absorption principle: λ⊗ π ∼= λ(dimπ).

Positivity: Jordan decomp.: M(G ) = spanM+(G );Polarization & GNS: B(G ) = spanP(G ) (positive definite func’s).

G abelian: B(G ) = M(G ) (Bochner); A(G ) = L1(G ) (Plancherel).

Philosophy:

Properties of L1(G ) ! properties of A(G );

properties of M(G ) ! properties of B(G ).

A property: operator amenability

Definition [Johnson ‘73, Ruan ‘95]

A Banach algebra, with compatible operator space structure (e.g.predual of HvN alg., or any operator algebra). A isoperator amenable if there is a bounded net

(dα) ⊂ A⊗A (op. proj. tens. prod.)

s.t.(a⊗ 1)dα − dα(1⊗ a)→ 0 and Π(dα) bdd. approx. id.

where Π(a⊗ b) = ab.

Note: L1(G )⊗L1(G ) ∼= L1(G × G ) [Grothendieck ‘50s]A(G )⊗A(G ) ∼= A(G × G ) [Effros-Ruan ‘90s]

If A ⊆ B(H), A⊗A → A⊗h A, Π factors accordingly.

Amenability: find bounded net (dα) ⊂ A⊗γ A (proj. t.p.) instead.

Operator amenability = “averagability”

W is A-bimodule, so is W∗: bϕa(w) = ϕ(awb).

Proposition, origins in [Helemskiı ‘80s]

Assume: A operator amenable; V ⊆ W c.b. A-bimodules, s.t.∃ c.b. linear projection P :W → V (V c.c. in W).Then ∃ c.b. E : V∗ →W∗, (Eψ)|V = ψ, E (aψ) = aE (ψ)

Idea: dα ≈∑nα

k=1 aα,k ⊗ bα,k . P∗α(ϕ) :=∑nα

k=1 aα,kP∗(bα,kϕ).

E w* cluster point of (P∗α) in CB(V∗,W∗) ∼= (V∗⊗W)∗. �Moral: P∗ was averaged to an A-module map.

Corollary, e.g. [Curtis-Loy ‘89]

I is a c.c. ideal in op. amenable A ⇒ I has b.a.i. (& op. amen.)

Idea: E ∗ : A∗∗ → I∗∗, E ∗(left unit) is a left unit ⇒ b.a.i. �

E.g.: A(G ) is c.c. in B(G ), via central proj’n in W∗(G ) ∼= B(G )∗.

On operator amenability of operator algebras

Note. [Johnson ‘73] (op.) amenability ⇔ certain derivations inner.

Theorem [Connes ‘78, Haagerup ‘83] ([Ruan ‘95])

C*-algebra: (op.) amenable ⇔ nuclear.

Theorem [Marcoux-Popov ‘16]

(op.) amenable commutative op. alg. ⇒ similar to C*-alg.

Corollary ([Scheinberg ‘77])

(op.) amenable uniform alg. is C(X )

Theorem [Choi-Farah-Ozawa ‘14]

∃ (op.) amenable op. algebra not isomorphic to C*-algebra.

Back to group and Fourier algebras

Philosophy: Properties of L1(G ) ! properties of A(G ).

Theorem

TFAE:(i) G is an amenable group;(ii) [Johnson ‘73] L1(G ) is an (op.) amenable Banach algebra;(iii) [Ruan ‘95] A(G ) is an op. amenable Banach algebra.

L1(G ) = L∞(G )∗ ⇒ “amenable” = “op. amenable”.

Operator spaces are essential to the above characterization:

Theorem [Forrest-Runde ‘04, Lau-Loy-Willis ‘96]

A(G ) amenable ⇔ G virtually abelian.

(Operator) weak amenability

Definition. A (op.) weakly amenable if every (c.) bdd. derivationD : A → A∗ is inner: D(a) = aϕ− ϕa.

[Bade-Curtis-Dales ‘87] A commutative: A (op.) weakly amen.⇔ 6 ∃ non-zero (c.) bdd. derivations to symmetric (c.c.) modules.

Theorem

(i) [Johnson ‘91] L1(G ) always (op.) weakly amenable.(ii) [Spronk ‘02, Samei ‘05] A(G ) always op. weakly amenble.

Theorem [Losert ‘19, Forrest-Runde ‘04]

A(G ) weakly amenable ⇔ Ge is abelian.

Lie case (⇒): [Choi-Ghandehari ‘15, Lee-Ludwig-Samei-S. ‘16].

Onto measure and Fourier-Stieltjes algebras

Philosophy: Properties of M(G ) ! properties of B(G ).

Theorem [Dales-Ghahramani-Helemskiı ‘02]

(i) M(G ) (op.) weakly amenable ⇔ G discrete.(ii) M(G ) (op.) amenable ⇔ G discrete and amenable.

M(G ) = C0(G )∗ ⇒ “amenable” = “op. amenable”.

Corollary

B(G ) amenable ⇔ G compact and virtually abelian.

Idea: B(G ) amen. ⇒ A(G ) amen. ⇒ G has cofinite abelian HM(H) ∼= B(H) = B(G )|H amenable ⇒ H compact. �

Naıve conjecture

B(G ) op. (weakly) amenable ⇔ G compact.

Naıve conjecture is wrong

Theorem [Runde-S. ‘04]

B(Qp oO×p ) is operator amenable and weakly amenable.

Questions

(1) When is B(G ) operator amenable?(2) When is B(G ) (operator) weakly amenable?(3) Can we answer (1) for any specific classes? Connected groups?

Rest of talk: focus on operator amenability of B(G ).

New idea: unitarizable topologies and projections

Tu(G ) = {τ ⊆ τG : (G , τ) top’l group and τ = σ(G ,Pτ )}where Pτ = {τ -cts. pos. def. func’s}.

E.g. (i) τG = σ(G ,P ∩A(G ))(ii) τap = σ(G ,Pfin), Pfin = {〈π(·)ξ, ξ〉 : π f.d., ξ ∈ Hπ}.

Often not Hausdorff, e.g. G = SLn(R),Rn o SO(n),Qp oO×p .

$τ =⊕

u∈Pτ πu (G.N.S.), τ = σ(G , {$τ})G$τ = $τ (G )

w .o.– semitopological semigroup

Gτ = U(H$τ ) ∩ G$τ – complete w.r.t. 2-sided uniformity

τ1 ⊆ τ2: $τ1 : (G , τ2)→ Gτ1 is uniformly continuous, henceuniquely extends to continuous ητ2τ1 : Gτ2 → Gτ1 with dense range.

Lattice operations and central projections

τ1, τ2 ∈ Tu(G ). τ1 ∨ τ2 = σ(G , {$τ1 ⊕$τ2})τ1 ∩ τ2 = σ(G ,Pτ1 ∩ Pτ2) = σ(G ,Pτ1∩τ2)– complete lattice

$ = $τG, G$ = $(G )

w .o.

ZE(G$) = {p ∈ G$ : p2 = p, p$(s) = $(s)p ∀s ∈ G}.

p1, p2 ∈ ZE(G$). p1p2 ∈ ZE(G$), so p1 ≤ p2 if p1p2 = p1.p1 ∨ p2 =

∏{p ∈ ZE(G$) : p1 ≤ p, p2 ≤ p}.

– complete lattice

A Galois connection

Theorem [S. ‘18]

(i) ∃ mapsP : Tu(G )→ ZE(G$), P(τ1) ≤ P(τ2) if τ1 ⊆ τ2T : ZE(G$)→ Tu(G ), T (p1) ⊆ T (p2) if p1 ≤ p2

P ◦ T = idZE(G$)

τ ⊆c T ◦ P(τ) : ηT◦P(τ)τ open, ker ηT◦P(τ)

τ compact.

(ii) Tu(G ) = T ◦ P(Tu(G )) complete sublattice [Tarski’s F.P.T.](iii) If τ ∈ Tu(G ) then

Bτ := P(τ) · B(G ) = {u ∈ B(G ) : τ -continuous}Iτ := (I − P(τ)) · B(G )C B(G )

so B(G ) = Bτ ⊕`1 Iτ (♥)

[Bouziad-Lamanczyk-Mentzen ‘01] Tu(Z) = {τap, τd} ∪ U|U| ≥ c (may be 2c), Zτ not l.c. for τ ∈ U

Aside: decomposition of representations

Unitary rep’n π : G → U(H), let π′′ : W∗(G )→ π(G )′′ ⊆ B(H).

Theorem [S. ‘18]

If τ ∈ Tu(G ) then get π-invariant subspaces

Hτπ = π′′(P(τ))H = {ξ ∈ H : π(·)ξ, τ−cts.}

Hτ,⊥π = (I − π′′(P(τ)))H = {ξ ∈ H : 0 ∈ π(W )ξw, ∀ eG ∈W ∈ τ}

With τ = τap get:

Corollary [Jacobs ‘54, Dye ‘65, Bergelson-Rosenblatt ‘88]

Hretπ = {ξ ∈ H : ξ ∈ π(G )η

w, ∀ η ∈ π(G )ξ

w} − “return”

Hwmπ = {ξ ∈ H : 0 ∈ π(G )ξ

w} − “weakly mixing”

= {ξ ∈ H : 0 ∈ π(W )ξw, ∀ eG ∈W ∈ τap}

First application to operator amenability

Theorem

B(G ) op. amenable ⇒ |Tu(G )| = |ZE(G$)| <∞.

Idea. If L ∈ Tu(G ) \ {τG} is a ∩-semilattice.Use (♥) to decompose

B(G ) = `1-⊕χ∈L

Aχ, graded: Aχ1Aχ2 ⊆ Aχ1χ2

where L = { semicharacters from L into {0, 1}}.Get complete quotient homomorphism Q : B(G )→ `1(L).If |Tu(G )| =∞, taking L↗ Tu(G ) \ {τG} shows non-op.amenability [Ghandehari-Hatami-S. ‘09, Duncan-Namioka ‘78]. �

E.g. G = Qp oO×p ,Rn o SO(n): Tu(G ) = {τap, τG}.

Case of connected groups

[Ruppert ‘84] G connected ⇒ ZE(G$) = E(G$)

Theorem [Mayer ‘97]

G connected:

|Tu(G )| = |E(G$)| <∞ ⇒ G totally minimal (∗)⇒ G/M = N o R

where – M certain co-Lie compact normal subgroup,– N nilpotent connected,– R connected linear reductive, and– R y N, only fixed point is e.

(∗) ∀ closed N C G , G/N has unique Hausdorff τ ⊆ τG/N .

Second application to operator amenability

Theorem

G connected: B(G ) operator amenable ⇔ G is compact.

Idea. B(G ) amenable ⇒ A(G ) has b.a.i.⇒ G amenable [Leptin ‘68].

Since, also, G/M = N o R amenable, R = K compact.

Assume G = N o K . (Removes many technicalities.)

Let V ⊆ N be in last non-trivial part of descending central series,be K -invariant and of minimal dimension ≥ 1.Notice V ⊆ Z (N), so V C G . V ∼= Rk as K y V w. no f.p.

Idea ... continued

[Cowling-Rodway ‘79] B(G )|V = BK (V ) op. amenable, whereBK (V ) = {u ∈ B(V ) | k 7→ u(k•) = u · k : K → B(V ) cts.}.

⇒ BK0 (V ) = BK (V ) ∩ C0(V ), c.c. ideal ⇒ has b.a.i.

⇒ B0(V )K = {u ∈ B0(V ) | u · k = u ∀k ∈ K}averaged from BK

0 (V ) via an expectation map ⇒ admits b.a.i.

B0(V )K ∼= M0(V )K – via F.S. transform

[Ragozin ‘73] [Mc (V )K ]∗ dimV ⊆ L1(V ), so Mc(V )K = M0(V )K

but L1(V )K ( Mc (V )K . �

Some odds and ends

Corollary

G almost connected: B(G ) operator amenable ⇔ G is compact.

[Liukkonen-Mislove ‘75] B(G )|Ge = B(Ge).

Remark. Question of op. amenability of B(G ) reduced to amenablegroups having:– compact connected components of identity, and– no non-compact central, nor open, abelian subgroups.

Question. G discrete: B(G ) op. amen. ⇒ |G | <∞?

Question. Op. weak amenability of B(G )? For connected G?

Preprints on arXiv

S., Weakly almost periodic topologies, idempotents and ideals,arXiv:1805.09892

S., On operator amenability of Fourier-Stieltjes algebras,arXiv:1806.08421

Thank-you!