groups Uncertainty principles for locally compact quantum Journal of Functional Analysis

Contents lists available atScienceDirect

Journal of Functional Analysis

www.elsevier.com/locate/jfa

Uncertainty principles for locally compact quantum groups

Chunlan Jiang^a,Zhengwei Liu^b,Jinsong Wu^c,∗

aDepartmentofMathematics,HebeiNormalUniversity,Shijiazhuang,Hebei 050024,PRChina

bDepartmentofMathematicsandDepartmentofPhysics,HarvardUniversity, Cambridge02138,USA

cInstituteforAdvancedStudyinMathematics,HarbinInstituteofTechnology, Harbin,Heilongjiang150001,PRChina

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received14June2017 Accepted15September2017 Availableonline22September2017 CommunicatedbyStefaanVaes

Keywords:

Uncertaintyprinciple

Locallycompactquantumgroup Group-likeprojection

Fouriertransform

In this paper, we prove the Donoho–Stark uncertainty principleforlocallycompactquantumgroupsandcharacterize the minimizer which are bi-shifts of group-like projections.

WealsoprovetheHirschman–Beckner uncertainty principle forcompact quantumgroupsanddiscretequantumgroups.

Furthermore, we show Hardy’s uncertainty principle for locallycompactquantumgroupsintermsofbi-shiftsofgroup- likeprojections.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

Uncertaintyprincipleswerefirststudiedinquantum mechanics and thenwidely de- velopedinharmonicanalysis,informationtheory, andquantum informationetc. In[9], DonohoandStark provedasupport-versionuncertaintyprinciplefor cyclicgroupsand applied it in signal recovery. Later Candes, Romberg, and Tao [7] developed this un-

* Correspondingauthor.

E-mailaddresses:[email protected](C. Jiang),[email protected](Z. Liu), [email protected](J. Wu).

https://doi.org/10.1016/j.jfa.2017.09.010

0022-1236/©2017ElsevierInc.Allrightsreserved.

certainty principleinthe theoryofcompressed sensing.TheDonoho–Starkuncertainty principle was proved forfinite abelian groups[25], locally compact abelian groups[23], compact groups [1], Kac algebras [6,20]. The minimizers of the Donoho–Stark uncer- tainty principle for locally compact abelian groups [9,25,23] werethe translations and modulationsofcharacteristicfunctionsofcompactopensubgroups.Fornoncommutative case,theauthors[14]showedtheDonoho–Starkuncertaintyprincipleforsubfactorsand introduced bi-shifts of biprojections for subfactors which generalized the modulations and translations ofcharacteristic functions ofsubgroups. The authorsshowed thatthe minimizers of the uncertaintyprinciple are bi-shifts of biprojections. For infinite case, LiuandWu[20]characterizedtheminimizersoftheDonoho–Starkuncertaintyprinciple forKacalgebras withbiprojections.

Hirschman uncertainty principle in terms of entropies was first introduced by Hirschman in [13]. In [2], Beckner proved the uncertainty principle with sharp con- stant for the real line R. The Hirschman–Beckner uncertainty principle generalized Heisenberg’s uncertainty principle in quantum mechanics. This uncertainty principle wasstudiedforlocallycompactabeliangroups[23],Kacalgebras [6,20],andsubfactors [14].TheminimizersoftheHirschman–Beckneruncertaintyprinciplewerecharacterized in[23]and [14].

Hardy’suncertaintyprincipleforRwasprovedin[11].Hardy’suncertaintyprinciples forarbitrarylocallycompactgroupwerestudiedrarely.In[14],theauthorsshowedthat Hardy’suncertaintyprincipleforsubfactorsbyusingtheminimizersoftheDonoho–Stark and theHirschman–Beckner uncertaintyprinciple.In[20], Liuand Wuproved Hardy’s uncertainty principle for Kac algebras with biprojections. Note that the authors [14]

showedthatthereareeightformsofabi-shiftofabiprojectionandHardy’suncertainty principlein[14,20]impliesthattheuniquenessofabi-shiftofabiprojection.

Locally compactquantum groupsintroducedby Kustermannsand Vaes[16,17]gen- eralized locally compact groupsand their duals. Compact quantum groupsintroduced byWoronowicz[27–30]arelocallycompactquantumgroups.Inthispaper,weprovethe Donoho–Starkuncertaintyprincipleforlocallycompactquantumgroupsandcharacter- izetheminimizersoftheuncertaintyprinciple.Weintroducethenotionofabi-shiftofa group-likeprojectionandshowthattheminimizersarebi-shiftsofgroup-likeprojections.

Forfinite abeliangroups,bi-shiftsofgroup-likeprojectionsarewavepackets[10].Wave packets arewidelyusedinquantum mechanics,information theory,etc.

Main Theorem 1 (Donoho–Stark uncertainty principle, Theorem 4.2, Proposition 4.7, Proposition 6.5). Suppose G is a locally compact quantum group. Then for any ω in L¹(G)∩L²(G),1≤t≤2,2≤s≤ ∞,wehave

Sr(ξt(ω))Sr(ι^s(λ(ω)))≥1.

Moreover thefollowingare equivalent:

1. ω∈L¹(G)∩L²(G)isaminimizer oftheDonoho–Stark uncertaintyprinciple.

2. ω is an extremal bi-partial isometry such that |ω|σ_t^ϕ = |ω|, σˆt(|λ(ω)|) = |λ(ω)|,

∀t∈R.

3. ω is a bi-partial isometry, |ω|σ^ϕ_t = |ω|, ∀t ∈ R, and λ(ω) is in L¹( ˆG) such that λ(λ(ω) ˆˆ ϕ)∞=λ(ω) ˆϕ.

4. ω∈L¹(G)∩L²(G)satisfies thatSr(ω)Sr(λ(ω))= 1 andσˆt(|λ(ω)|)=|λ(ω)|. 5. ω∈L¹(G)∩L²(G)satisfies thatSr(ω)Sr(λ(ω))= 1 andSr(ξ(ω))Sr( ˆΛ(λ(ω)))= 1.

6. ω isabi-shiftof agroup-likeprojectionB∈L¹(G).

Notethat Sr(x)istheϕ-valueofthesupport projectionof x,anditwill be explainedin Section3.

The Donoho–Stark uncertainty principle for locally compact quantum groups is a series ofinequalities which isdifferent from the casefor unimodularKac algebras.For a locally compact quantum group G, we define an entropy H(ξ) of ξ in L²(G). Then weprovedtheHirschman–Beckneruncertaintyprincipleforcompactquantumgroupsor discretequantumgroups.

MainTheorem2(Hirschman–Beckneruncertaintyprinciple,Theorem 5.15).SupposeG isa compactquantum group or adiscrete quantum group andφ=ϕ(Jϕ·Jϕ).Let ω ∈ L¹(G)∩L²(G) such that ξ(ω)= 1. If H(ξ(ω)), H( ˆΛ(λ(ω))), |logd|Jϕξ(ω),Jϕξ(ω) |log ˆd|JˆΛ(λ(ω)),ˆ JˆΛ(λ(ω))ˆ are finite,then

H(ξ(ω)) +H( ˆΛ(λ(ω)))≥ − (logd)Jϕξ(ω), Jϕξ(ω) − (log ˆd) ˆJΛ(λ(ω)),ˆ JˆΛ(λ(ω))ˆ . Moreover, for any ξ ∈ L²(G) if H(ξ), H(F2(ξ)), |logd|Jϕξ,Jϕξ, |log ˆd|JˆF2(ξ), JˆF2(ξ)are finite,then

H(ξ) +H(F2(ξ))≥ − (logd)Jϕξ, Jϕξ − (log ˆd) ˆJF2(ξ),JˆF2(ξ).

Byusingbi-shiftsofgroup-likeprojections,weshowHardy’suncertaintyprinciplefor locallycompactquantum groupswithgroup-likeprojections.

Main Theorem 3(Hardy’s uncertainty principle, Theorem 6.6). Suppose G isa locally compact quantum group with a bi-shift w of a group-like projection. Let x ∈ L¹(G)∩ L^∞(G)be suchthat

|x^∗| ≤C|w^∗|, |λ(xϕ)| ≤C|λ(wϕ)|, forsomeC,C>0.Thenxisamultiple of w.

Thetechniquesin[4,12,16]forlocallycompactquantumgroupsandnoncommutative L^tspaceswillbefrequentlyusedinthepaper.Thispaperisorganizedasfollows.InSec- tion2, wegooverthedefinitionandsomebasicpropertiesoflocallycompact quantum

groups and noncommutativeL^t spaces for 1≤t ≤ ∞. InSection3, we study support projectionsorrangeprojectionsofelementsinnoncommutativeL^tspaces.InSection4, we show the Donoho–Stark uncertainty principle for locally compact quantum groups and obtainthepropertiesoftheminimizersoftheDonoho–Starkuncertaintyprinciple.

In Section 5, we study the derivative of the t-norm and show the Hirschman–Beckner uncertaintyprinciplefor compactquantum groupsor discretequantum groups.InSec- tion 6, we proveHardy’suncertaintyprinciple forlocally compact quantum groups.In Section 7, we obtain more results on Young’s inequality for locally compact quantum groups.

2. Preliminaries

In this section,we will recall somepropertiesof noncommutativeL^t spaces and the definitionandpropertiesoflocally compactquantumgroups.

LetMbe avon Neumannalgebrawithanormalsemi-finitefaithfulweightϕand Nϕ={x∈ M:ϕ(x^∗x)<∞}, Mϕ=N^∗_ϕNϕ.

Denote by ∇ϕ, Jϕ, σ^ϕ the modular operator, modular conjugation and modular au- tomorphism group associated with ϕ. It is known that ϕ is invariant under σ^ϕ, i.e.

ϕσ^ϕ_t = ϕ for any t ∈ R. For any x ∈ D(σ^ϕ_i/2), the domain of σ_i/2^ϕ , we have that ϕ(x^∗x)=ϕ(σ_i/2^ϕ (x)σ_i/2^ϕ (x)^∗).Denote byM⁺_ϕ theset ofallpositiveelementsinMϕ.

Let (Hϕ,π_ϕ,Λ_ϕ) be the Gelfand–Naimark–Segal (GNS) semi-cyclic representation, where Λϕ : Nϕ → Hϕ is an inclusionmap. We assume thatM act on Hϕ. Therefore wewillomitπϕ.ThemodularconjugationJϕsatisfiesthatJϕΛϕ(x)= Λϕ(σ_i/2^ϕ (x)^∗) for any x ∈ Nϕ∩ D(σ_i/2^ϕ ). For any x ∈ Mϕ and a ∈ D(σ₋^ϕ_i), we have ax,xa ∈ Mϕ and ϕ(ax)=ϕ(xσ₋i(a)).

Denote by M∗ thepredual ofM. Denote byM⁺_∗ the set ofallpositivelinearfunc- tionals inM∗.Foranyω∈ M∗,thelinearfunctional ω∈ M∗ isgivenbyω(x)=ω(x^∗) foranyxinM.GivenxinMandω∈ M∗,thelinearfunctionalsxωandωxaregiven by(xω)(y)=ω(yx) and(ωx)(y)=ω(xy) foranyy inM.

TheTomitaalgebraTϕ isgivenby

Tϕ={x∈ M:xis analytic w. r. t.σ^ϕ andσ_z^ϕ(x)∈N^∗_ϕ∩Nϕ,∀z∈C}. It isknownthatTϕ,Tϕ² areσ-stronglydenseinM.

LetLϕ={x∈N_ϕ:xϕ∈ M∗}andRϕ={x∈N^∗_ϕ:ϕx∈ M∗}.Bytheresultsin[3], we havethatTϕ²⊂ Lϕand Rϕ=L^∗ϕ.

ThenoncommutativeL^tspaceL^t(M) isthecomplexinterpolationspace(M,M∗)_[1/t]

of M∗ and M for 1 ≤ t ≤ ∞. In [3], L^t(M) is written as L^t(M)_left or L^t_(−1/2)(M).

Note that L¹(M) = M∗, L^∞(M) = M, and L¹(M)∩L^∞(M) = Lϕ. Denote by ι^t:Lϕ→L^t(M) theembeddingofLϕinto L^t(M). Itisknownthatι^t(Lϕ) is densein L^t(M) forany1≤t≤ ∞.Byresultsin[3],weidentifyL²(M) withHϕ.Moreover,

L¹(M)∩L²(M) =I={ω∈ M∗|Λϕ(x)→ω(x^∗), x∈Nϕ is a bounded map.}

=M∗∩ Hϕ, and

L²(M)∩L^∞(M) =Nϕ=Hϕ∩ M.

Denotebyξt:I →L^t(M) for1≤t≤2 theembeddingfromI toL^t(M).Fort= 2, we useξ(ω) insteadof ξ₂(ω) wheneverω∈ I. Denote byι^s:N_ϕ →L^s(M) for2≤s≤ ∞ theembeddingfrom Nϕ toL^s(M).For s= 2, weuse Λ_ϕ(x) instead ofι²(x) whenever x∈Nϕ.Notethatweusethesamenotationas ι^t:Lϕ→L^t(M) here,sincethere isno confusion.

LetφbeafixednormalsemifinitefaithfulweightonthecommutantM ofMonHϕ. AcloseddenselydefinedoperatoraontheHilbertspaceHϕisγ-homogeneouswithγ∈R ifxa⊆aσ_iγ^φ(x) forallx∈ M analyticwithrespecttothemodularautomorphismσ^φ of M.TheHilsum L^tspaceL^t(φ) isthespaceofallcloseddenselydefinedoperatorsaon Hϕsuchthatifx=u|x|isthepolardecomposition,then|x|^tisthespatialderivativeofa positivelinearfunctionalω∈ M∗andu∈ M.NotethatL^∞(φ)=M.Thedistinguished spatial derivative d = ^dϕ_dφ is a strictly positive self-adjoint operator acting on Hϕ and σ_t^ϕ(x)=d^itxd^−itforeveryxinM;σ^φ_t(y)=d^−ityd^itforeveryy inM.Throughoutthe paper, wewill identifystrongproduct,strongsum asproduct,sumrespectively. There isanisometricisomorphismΦ_t:L^t(φ)→L^t(M) for1≤t≤ ∞suchthat

Φ_t(xd^1/t) =ι^t(x), x∈ Tϕ². Ift≥2,wehavethatΦt(xd^1/t)=ι^t(x) forany x∈Nϕ.

ByTheorem 2.4in[5],forany x∈ Tϕ,t∈[2,∞],wehavexd^1/t=d^1/tσ_i/t^ϕ (x).

Nowwewillrecallthedefinitionoflocallycompactquantumgroups.Alocallycompact quantumgroupG= (M,Δ,ϕ,ψ) consistsof

(1) AvonNeumannalgebraM;

(2) Aunitalnormal*-homomorphismΔ:M→ M⊗Msatisfying(Δ⊗ι)Δ= (ι⊗Δ)Δ, whereι:M→ Mistheidentity.

(3) Twonormalsemifinitefaithfulweightϕ,ψ onMsuchthat ϕ((ω⊗ι)Δ(x)) =ϕ(x)ω(1), ∀ω∈ M⁺_∗, x∈M⁺_ϕ, ψ((ι⊗ω)Δ(x)) =ψ(x)ω(1), ∀ω∈ M⁺_∗, x∈M⁺_ψ, ϕistheleftHaar weightandψ istherightHaarweight.

Suppose Hϕ is the Hilbert space arising from the GNS representation of M with respectto ϕand assumethatMactsonHϕ.

ThemultiplicativeunitaryoperatorW ∈ B(Hϕ⊗ Hϕ) isdefinedby W^∗(Λϕ(x)⊗Λϕ(y)) = (Λϕ⊗Λϕ)(Δ(y)(x⊗1)) forany x,y∈Nϕ. WehaveΔ(x)=W^∗(1⊗x)W foranyx∈ M.

There isadual locallycompactquantum groupGˆ = ( ˆM,Δ,ˆ ϕ,ˆ ψ).ˆ Recallthat Mˆ ={(ω⊗ι)(W)|ω∈ B(H)_∗}^{σ-strong-∗},

Wˆ = ΣW^∗Σ, where Σ is the flip on Hϕ⊗ Hϕ. Δ(x)ˆ = ˆW^∗(1⊗x) ˆW for any x∈ Mˆ. The Fourier representation λ: M∗ → Mˆ is given byλ(ω)= (ω⊗ι)(W) forany ω in M∗. Thedualleft HaarweightϕˆistheuniquenormalsemifinitefaithfulweightonMˆ having (Hϕ,ι,Λ) asˆ aGNS-constructionwhere Λ isˆ alinearmap from Nϕˆ to Hϕ given byΛ(λ(ω))ˆ =ξ(ω) forany ω∈L¹(M)∩L²(M).There aretheantipodeS,theunitary antipodeR,andascalingautomorphismτonM.TheleftandtherightHaarweightsϕ, ψ satisfies ψ(·)=ϕ(δ^1/2·δ^1/2), where δis themodularelement ofthe locallycompact quantum groupG. Note thatδ is a uniquestrictly positive element affiliatedwith M and

Δ(δ) =δ⊗δ, R(δ) =δ⁻¹, τt(δ) =δ, ∀t∈R.

Therearenormcontinuousone-parameterrepresentationsρ,δ^∗,τ^∗ofRonM∗given by

ρt(ω)(x) =ω(δ^−itτ₋t(x)), δ_t^∗(ω)(x) =ω(δ^itx), τ_t^∗(ω)(x) =ω(τt(x))

respectively forall ω ∈ M∗, x∈ Mand t∈R. Forthe dual locally compactquantum groupGˆ,therearenorm continuousone-parametergroupsσ,ˆ ˆτ onMˆ suchthat

ˆ

σt(λ(ω)) =λ(ρt(ω)), τˆt(λ(ω)) =λ(ωτ_−t)

respectivelyforallt∈Randω∈ M∗.ThereistheunitaryantipodeRˆonMˆ suchthat R(λ(ω))ˆ =λ(ωR) forallω∈ M∗.Thereisalso anantipodeSˆ= ˆRˆτ_−i/2 onMˆ.Denote byJˆthemodularconjugationassociatedto ϕ.ˆ

Recall thattheantipodeS=Rτ_−i/2 onMhasthefollowingproperties:

1. S(ψ⊗ι)((x^∗⊗1)Δ(y))= (ψ⊗ι)(Δ(x^∗)(y⊗1)), ∀x,y∈Nψ. 2. S((ι⊗ϕ)(Δ(x^∗)(1⊗y)))= (ι⊗ϕ)((1⊗x^∗)Δ(y)), ∀x,y∈Nϕ. 3. S((ι⊗ω)(W))= (ι⊗ω)(W^∗), ∀ω∈ B(H)_∗.

Denote byν thescalingconstantofthelocallycompactquantum groupG.Then ϕτt=ν^−tϕ, ϕσ^ψ_t =ν^tϕ, ∀t∈R.

SomefundamentalcommutationrelationsforthelocallycompactquantumgroupGare Δσ_t^ϕ= (τt⊗σ^ϕ_t)Δ, Δτt= (τt⊗τt)Δ.

For a locally compact quantum group G= (M,Δ,ϕ,ψ) and t ∈ [1,∞], we denote by L^t(G) the complex interpolation space L^t(M). For any ω ∈ L¹(G)∩L²(G), the L^t-FouriertransformFt:L^t(G)→L^s( ˆG),1/t+ 1/t = 1 isgivenby

Ft(ξ_t(ω)) =ι^t(λ(ω)).

BytheHausdorff–Younginequality[5,3],wehavethatFt≤1.

SupposeGisalocallycompactquantumgroup.AprojectionB inL^∞(G) isagroup- like projectionifΔ(B)(1⊗B)=B⊗B and B= 0.A projectionB inL¹(G)∩L^∞(G) is abiprojection if λ(Bϕ) isamultiple of aprojection. For more detailson group-like projections, we refer to [18]. In [19], Liu, Wangand Wu show that abiprojection is a group-likeprojectioninL¹(G) ifϕ=ψor ϕistracial.

Intheendofthesection,werecalltheresolventconvergenceforunboundedself-adjoint operators, let an,n = 1,. . .and a be (unbounded) self-adjoint operators on a Hilbert spaceH. Theelements an issaid toconverge to ainthe strongresolvent sense if(z− an)⁻¹ → (z−a)⁻¹ inthe strong operatortopology for all z ∈ C with z = 0.an is saidtoconverge toaintheweakresolventsenseif(z−an)⁻¹→(z−a)⁻¹ intheweak operatortopology for all z ∈ Cwith z = 0. Theweak resolvent convergence implies thestrongresolventconvergence.Formoredetails,wereferthereadersto[24].

3. Supportprojections

Inthissection,wedefinetherangeprojectionsandthesupportprojectionsofelements in the noncommutative L^t space L^t(M) with the help of the Hilsum L^t spaces and investigatetheirproperties,whereMisavonNeumannalgebrawithanormalsemifinite faithfulweightϕ.

Forany xin L^t(M), 1≤t ≤ ∞,we define the rangeprojection Rl(x) (which is on thelefthandside)ofxtobetherangeprojectionR(Φ⁻¹_t (x)) ofΦ⁻¹_t (x) andthesupport projectionRr(x) (whichisontherighthandside)ofxtobeR(Φ⁻¹_t (x)^∗).Notethatby thedefinitionoftheHilsumL^tspace,weseethatR(Φ⁻¹_t (x)),R(Φ⁻¹_t (x)^∗)∈ Mforany x∈L^t(M).We denotebySl(x)=ϕ(Rl(x)) andSr(x)=ϕ(Rr(x)).Ift=∞,then for anyx∈L^∞(M)(=M),wehavethat

Rl(x) =R(x), Rr(x) =R(x^∗).

Forany ω inL¹(M)(=M∗), wedenote thesupport projectionof ω byR(ω) which isgivenby

R(ω) = 1−p,

where pistheunionofallprojectionpα inMsuchthatω(pαx)= 0,∀x∈ M.

ForanyξinL²(M)(=Hϕ),wedenotethesupportprojectionofξbyR(ξ) givenby R(ξ) = 1−p,

where pis theunionof allprojectionp_αinM suchthatp_αξ= 0. Forany x∈ Mand ξ∈L²(M),therightactionofxonξ isgivenbyξx=J_ϕx^∗J_ϕξ.

Lemma 3.1. Suppose M is a von Neumann algebra with a normal semifinite faithful weightϕ.Thenforanyx∈ M,ω ∈L¹(M),ξ∈L²(M),wehavethat

xΦ⁻¹₁ (ω) = Φ⁻¹₁ (xω), Φ⁻¹₁ (ω)x= Φ⁻¹₁ (ωx), and

xΦ⁻₂¹(ξ) = Φ⁻₂¹(xξ), Φ⁻₂¹(ξ)x= Φ⁻₂¹(ξx).

Proof. Supposethatx∈ Tϕ²andassumethat{xβ}βisanetinTϕ²suchthatlimβxβϕ− ω= 0.Then

xΦ⁻¹₁ (ω)−Φ⁻¹₁ (xω)1,φ= lim

β xΦ⁻¹₁ (ω)−xx_βd+xx_βd−Φ⁻¹₁ (xω)1,φ

≤lim

β xΦ⁻¹₁ (ω)−xxβd1,φ+ lim

β xxβd−Φ⁻¹₁ (xω)1,φ

≤ x∞lim

β ω−x_βϕ+ lim

β xx_βϕ−xω= 0, i.e. xΦ⁻₁¹(ω)= Φ⁻₁¹(xω) foranyxinTϕ².

Supposethatx∈ Mandtakeaboundednet{xα}α⊆ Tϕ²suchthatxα convergesto x σ-strongly.LetΦ⁻¹₁ (ω)=w|Φ⁻¹₁ (ω)| be thepolardecompositionand ω0 thepositive normallinearfunctional suchthat|Φ⁻¹₁ (ω)|=^dω_dφ⁰.Then

xΦ⁻₁¹(ω)−Φ⁻₁¹(xω)1,φ= lim

α xΦ⁻₁¹(ω)−xαΦ⁻₁¹(ω) + Φ⁻₁¹(xαω)−Φ⁻₁¹(xω)1,φ

≤lim

α (x−x_α)Φ⁻₁¹(ω)1,φ+ lim

α x_αω−xω

≤lim

α (x−x_α)w|Φ⁻¹₁ (ω)|^1/22,φ|Φ⁻¹₁ (ω)|^1/22,φ

= lim

α ω0(w^∗(x−xα)^∗(x−xα)w)^1/2ω0(1)^1/2= 0,

i.e. xΦ⁻¹₁ (ω) = Φ⁻¹₁ (xω) for all x in M. The rest of the Lemma can be proved simi- larly. 2

Proposition3.2. SupposeMisavonNeumannalgebrawithanormalsemifinitefaithful weight ϕ.Thenforanyω inL¹(M),wehavethat

Rl(ω) =R(ω), Rr(ω) =R(ω);

forany ξin L²(M),wehave that

Rl(ξ) =R(ξ), Rr(ξ) =R(Jϕξ).

Proof. ThiscanbeobtainedfromLemma 3.1. 2

Proposition3.3.SupposeMisavonNeumannalgebra withanormal semifinitefaithful weight ϕ. Thenfor any ω ∈L¹(M)∩L²(M),y ∈L²(M)∩L^∞(M),x∈ M,we have that

xΦ⁻_t¹(ξt(ω)) = Φ⁻_t¹(ξt(xω)),1≤t≤2, xΦ⁻¹_s (ι^s(y)) = Φ⁻¹_s (ι^s(xy)),2≤s≤ ∞. Moreover,forany ω∈L¹(M)∩L²(M),y∈L²(M)∩L^∞(M),wehave

Rl(ω) =Rl(ξ_t(ω)),1≤t≤2, Rl(ι^s(y)) =Rl(y),2≤s≤ ∞.

Proof. ByResult8.6in[16],wehavethatxξ(ω)=ξ(xω) foranyx∈ M,ω∈L¹(M)∩ L²(M).Thatistosaythatxω∈L¹(M)∩L²(M).ByProposition3.4in[3],foragiven ω ∈ L¹(M)∩L²(M), there is a net {x_β}β ⊂ Tϕ² such that lim_βω−x_βϕ = 0 and lim_βξ(ω)−Λ_ϕ(x_β)= 0.Thenforany1≤t≤2,x∈ Tϕ²,wehavethat

xΦ⁻¹_t (ξ_t(ω))−Φ⁻¹_t (ξ_t(xω))t,φ

= lim

β xΦ⁻¹_t (ξt(ω))−xxβd^1/t+xxβd^1/t−Φ⁻¹_t (ξt(xω))t,φ

= lim

β xΦ⁻¹_t (ξt(ω))−xxβd^1/tt,φ+xxβd^1/t−Φ⁻¹_t (ξt(xω))t,φ

≤ x∞Φ⁻_t¹(ξt(ω))−xβd^1/tt,φ+ lim

β ξt(xxβϕ)−ξt(xω)t

≤lim

β xx_βϕ−xω^2/t−1Λ_ϕ(xx_β)−ξ(xω)^2−2/t

= 0,

i.e.xΦ⁻¹_t (ξ_t(ω))= Φ⁻¹_t (ξ_t(xω)) foranyxinTϕ².

For any x in M, there is abounded net {x_α}α ⊂ Tϕ² such that x_α converges to x σ-strongly.Then forany ω∈L¹(M)∩L²(M), letΦ⁻_t¹(ξt(ω))=wt|Φ⁻_t¹(ξt(ω))|be the polardecomposition,andwehave

xΦ⁻_t¹(ξt(ω))−Φ⁻_t¹(ξt(xω))t,φ

= lim

α xΦ⁻¹_t (ξ_t(ω))−x_αΦ⁻¹_t (ξ_t(ω)) + Φ⁻¹_t (ξ_t(x_αω))−Φ⁻¹_t (ξ_t(xω))t,φ

≤lim

α (x−x_α)w_t|Φ⁻¹_t (ξ_t(ω))|^t/2|Φ⁻¹_t (ξ_t(ω))|^1−t/2t,φ

+ lim

α xαω−xω^2/t−11 ξ(xαω)−ξ(xω)^2−2/t2

≤lim

α (x−x_α)w_t|Φ⁻¹_t (ξ_t(ω))|^t/22,φ|Φ⁻¹_t (ξ_t(ω))|^1−t/2(2−t)/2t

= 0

i.e. xΦ⁻¹_t (ξt(ω))= Φ⁻¹_t (ξt(xω)) foranyxinM.

Forthesecondequation,wecanuseasimilarargumentasaboveandProposition3.6 in [3] to see it. The remaining two equations are obtained directly from the first two equations. 2

NowwecandefinetheleftactionofMonL^t(M) forany1≤t≤ ∞.Theleftaction is givenas

xy= Φt(xΦ⁻_t¹(y)), ∀x∈ M, y∈L^t(M),1≤t≤ ∞. Similarly,wecandefinetherightactiononL^t(M) for1≤t≤ Mby

yx= Φt(Φ⁻¹_t (y)x), ∀x∈ M, y∈L^t(M),1≤t≤ ∞.

Proposition3.4. SupposeMisavonNeumannalgebrawithanormalsemifinitefaithful weight ϕ.Thenforanyω ∈L¹(M)∩L²(M),y∈L²(M)∩L^∞(M),x∈ Tϕ²,wehave

Φ⁻¹₂ (ξ(ω))x= Φ⁻¹₂ (ξ(ωσ^ϕ_i/2(x))), Φ⁻¹₂ (Λϕ(y))x= Φ⁻¹₂ (Λϕ(yσ^ϕ_−i/2(x))).

Proof. Foranyy∈Nϕ, wehavethat

ξ(ω)x,Λϕ(y)= Jϕx^∗Jϕξ(ω),Λϕ(y)

= ξ(ω), JϕxJϕΛϕ(y)

= ξ(ω),Λϕ(yσ_−i/2^ϕ (x^∗))

=ω(σ^ϕ_−i/2(x^∗)^∗y^∗)

=ω(σ^ϕ_i/2(x))(y^∗)

= ξ(ω(σ^ϕ_i/2(x)),Λϕ(y),

i.e. ξ(ω)x=ξ(ω(σ_i/2^ϕ (x))) forany x∈ Tϕ².ByLemma 3.1,we havethefirstequationis true.Similarly,wecanprovethesecondequation. 2

Generalizingtheresultsinthepropositionabove,wehavethefollowing proposition.

Proposition3.5. SupposeMisavonNeumannalgebrawithanormalsemifinitefaithful weight ϕ. Then for any ω ∈ L¹(M)∩L²(M), y ∈ L²(M)∩L^∞(M), 1 ≤ t ≤ 2, 2≤s≤ ∞,x∈ Tϕ²,wehave

Φ⁻_t¹(ξt(ω))x= Φ⁻_t¹(ξt(ωσ_i−i/t^ϕ (x))), Φ⁻¹_s (ι^s(y))x= Φ⁻¹_s (ι^s(yσ^ϕ_−i/s(x))) Proof. ByProposition3.4 in[3],there exists anet{x_β}β ⊂ Tϕ² suchthatlim_βx_βϕ− ω= 0 andlimβΛϕ(xβ)−ξ(ω)= 0. Thenwehave

Φ⁻_t¹(ξ_t(ω))x−Φ⁻_t¹(ξ_t(ωσ_i−i/t^ϕ (x)))t,φ

= lim

β Φ⁻¹_t (ξ_t(ω))x−x_βd^1/tx+x_βd^1/tx−Φ⁻¹_t (ξ_t(ωσ_i^ϕ_−i/t(x)))t,φ

≤lim

β Φ⁻_t¹(ξ_t(ω))x−x_βd^1/txt,φ+ lim

β x_βd^1/tx−Φ⁻_t¹(ξ_t(ωσ_i−i/t^ϕ (x)))t,φ

≤lim

β Φ⁻¹_t (ξt(ω))−xβd^1/tt,φx∞+ lim

β xβd^1/tx−Φ⁻¹_t (ξt(ωσ_i−i/t^ϕ (x)))t,φ

= lim

β x_βσ₋^ϕ_i/t(x)d^1/t−Φ⁻¹_t (ξ_t(ωσ_i^ϕ_−i/t(x)))t,φ

≤lim

β max{x_βσ_−i/t^ϕ (x)ϕ−ωσ^ϕ_i−i/t(x),Λ_ϕ(x_βσ_−i/t^ϕ (x))−ξ(ωσ_i−i/t^ϕ (x))}

= lim

β max{xβϕσ^ϕ_i−i/t(x)−ωσ_i−i/t^ϕ (x),Λϕ(xβ)σ_i/2−i/t^ϕ (x)−ξ(ω)σ^ϕ_i/2−i/t(x)}

= 0,

i.e.Φ⁻¹_t (ξt(ω))x= Φ⁻¹_t (ξt(ωσ_i^ϕ_−i/t(x))).Similarly,onecanshowthatthesecondequation istrue. 2

Forany ω ∈L¹(M),we letω =vω|ω|be the polardecomposition. ByTheorem 4.2 inChapter3of [26], wesee thatv^∗_ωv_ω=R(|ω|)=R(ω).Ifω ∈L¹(M)∩L²(M), then

|ω|∈L¹(M)∩L²(M).

Proposition3.6. LetG bealocally compactquantum group. Forany ω∈L¹(G),t∈R, wehave

R(ω◦σ^ϕ_t) =σ_−t^ϕ (R(ω)), R(ω◦τt) =τ₋t(R(ω)) R(δ^∗_t(ω)) =δ^−itR(ω)δ^it, R(ω◦R) =R(R(ω)) Proof. Weleavetheproofto thereader. 2

4. Donoho–Starkuncertaintyprinciple

Inthissection,weprovetheDonoho–Starkuncertaintyprincipleforlocally compact quantumgroupsand showaseriesofequivalent statementsforthecharacterizationsof minimizers of the uncertainty principle. Moreover, we obtain biprojections from mini- mizersoftheuncertaintyprinciple.

Proposition4.1. SupposeMisavonNeumannalgebrawithanormalsemifinitefaithful weight ϕ.Thenforanyω ∈L¹(M)∩L²(M),wehavethat

Sr(ξt(ω))≥ ω²

ξ(ω)², 1≤t≤2;

forany y∈L²(M)∩L^∞(M),we havethat Sr(ι^s(y))≥ Λ_ϕ(y)²

y²_∞ , 2≤s≤ ∞.

Proof. Fixsome1≤t≤2.Supposethatϕ(Rr(ξt(ω)))<∞.Then when1< t≤2,we have

ω= sup

x∞=1|ω(x)|

= sup

x∞=1,x∈Tϕ²

|ω(x)|

= sup

x∞=1,x∈Tϕ²

ξt(ω), xR^∗ϕ,Rϕ

= sup

x∞=1,x∈Tϕ²

Φ⁻_t¹(ξt(ω))d^t−1t xdφ

= sup

x∞=1,x∈Tϕ²

Φ⁻¹_t (ξ_t(ω))Rr(ξ_t(ω))d^t−1t xdφ

≤ sup

x∞=1,x∈Tϕ²

ξ_t(ω)tRr(ξ_t(ω))d^t−1t _t−1^t ,φx∞

=ξt(ω)tι^t−1t (Rr(ξt(ω)))_t−1^t

≤ ω^2t−1ξ(ω)^2−2tRr(ξ_t(ω))²⁻2 ^2tRr(ξ_t(ω))∞^2t−1

=ω^2t−1ξ(ω)^2−2tSr(ξ_t(ω))^1−1t i.e.

Sr(ξt(ω))≥ ω²

ξ(ω)², 1< t≤2.

Whent= 1,suppose ϕ(R(ω))<∞, wehave ω= sup

x^∗∈Nϕ,x∞=1

|ω(x)|

= sup

x^∗∈Nϕ,x∞=1

|ω(R(ω)x)|

= sup

x^∗∈Nϕ,x∞=1| ξ(ω),Λϕ(x^∗R(ω))|

≤ sup

x^∗∈Nϕ,x∞=1|ξ(ω)x^∗Λϕ(R(ω))

=ξ(ω)ϕ(R(ω))^1/2

=ξ(ω)Sr(ω)^1/2. Hence

Sr(ξ_t(ω))≥ ω²

ξ(ω)², 1≤t≤2.

Supposethatϕ(Rr(ι^s(y)))<∞for2< s≤ ∞. Thenwehave Λϕ(y)= sup

ξ=1| Λϕ(y), ξ|

= sup

x∈L²(φ),x2,φ=1|

yd^1/2xdφ|

= sup

x∈L²(φ),x2,φ=1|

yd^1/sRr(ι^s(y))d^12−1sxdφ|

≤ ysι^s−22s (Rr(ι^s(y)))_s−2^2s

≤ Λ_ϕ(y)^2sy¹⁻∞^2sΛ_ϕ(Rr(ι^s(y)))^s−2s Rr(ι^s(y))∞^s2

=Λϕ(y)^2sy¹∞^−2s(Sr(ι^s(y)))^s2s−2, i.e.

Sr(ι^s(y))≥Λϕ(y)²

y²_∞ , 2< s≤ ∞. Whens= 2,suppose ϕ(Rr(Λϕ(y)))<∞,wehave

Λϕ(y)=Λϕ(y)Rr(Λϕ(y))

=yJ_ϕΛ_ϕ(Rr(Λ_ϕ(y)))

≤ y∞Λϕ(Rr(Λϕ(y)))

=y∞ϕ(Rr(Λ_ϕ(y))^1/2

=y∞Sr(Λ_ϕ(y))^1/2. Hence

Sr(ι^s(y))≥ Λ_ϕ(y)²

y²_∞ , 2≤s≤ ∞. 2

Theorem4.2(Donoho–Starkuncertaintyprinciple).SupposeGisalocallycompactquan- tum group. Thenforany ω inL¹(G)∩L²(G),1≤t≤2,2≤s≤ ∞,we have

Sr(ξt(ω))Sr(ι^s(λ(ω)))≥1.

Proof. Supposethatϕ(R(ω))<∞andϕ(ˆR(λ(ω)^∗))<∞.ThenbyProposition 4.1,we have

Sr(ξt(ω))Sr(ι^s(λ(ω)))≥ ω²

ξ(ω)²Λ(λ(ω))ˆ ²

λ(ω)²_∞ = ω² λ(ω)²_∞ ≥1.

Hencewehavethetheoremproved. 2

Definition4.3.SupposeGisalocallycompactquantumgroup.Anelementω∈L¹(G)∩ L²(G) issaidtobeaminimizeroftheDonoho–StarkuncertaintyprincipleinTheorem 4.2 if

Sr(ξ_t(ω))Sr(ι^s(λ(ω))) = 1 forall1≤t≤2,2≤s≤ ∞.

Proposition 4.4. Suppose that M is a von Neumann algebra with a normal semifinite faithful weightϕ.Letω∈L¹(M)∩L²(M)be suchthat ϕ(Rr(ω))<∞.If

Sr(ω) = ω² ξ(ω)²,

then there is a partial isometry v ∈ L¹(M)∩L^∞(M) such that ω = μωvϕ for some μω>0andRr(ξt(ω))=Rr(ω)=σ_s^ϕ(Rr(ω))forany 1≤t≤2,s∈R.

Proof. Letω=v_ω|ω|bethepolardecomposition.Wehavethat ϕ(v_ω^∗v_ω) =ϕ(R(ω)) =ϕ(Rr(ω))<∞. Then

ω=ω(v^∗_ω)

= ξ(ω),Λ_ϕ(v_ω)

≤ ξ(ω)ϕ(R(ω))^1/2

=ω. BytheCauchy–Schwarzinequality,wehavethat

ξ(ω) =μωΛϕ(vω),