# CommutativeC⁎-Algebras of Toeplitz Operators via the Moment Map on the Polydisk

Authors: Mauricio Hernández-Marroquin; Armando Sánchez-Nungaray; Luis Alfredo Dupont-García
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We found that in the polydiskDn there exist (n+1)(n+2)/2 different classes of commutative C⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that the C⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutative C⁎-algebra of Toeplitz operators.


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## Body

## 1. Introduction

In recent years, Vasilevski, Quiroga, and coauthors found a connection that the commutativeC∗-algebras generated by Toeplitz operators acting on the weighted Bergman space were described and classified for the case of the unit disk and the unit ball in Cn; see [1, 2] for further results and details. The classification result states that given a maximal Abelian subgroup G of biholomorphisms of the unit ball or the unit disk (i.e., ball of dimension one), the C∗-algebra generated by Toeplitz operators whose symbols are invariant under the action of G is commutative on each weighted Bergman space.For the unit ball of dimensionn there are n+2 models of maximal Abelian subgroups G of biholomorphisms; all others are conjugated with one of them via biholomorphisms of the unit ball.We apply the above method to the polydiskDn and we find that there exist (n+1)(n+2)/2 different classes of commutative C∗-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. Moreover, we show that in each case the Toeplitz operator Ta is unitary equivalent to a multiplication operator γa,kI. The explicit form of γa,k is given.The main result of this paper is that we relate to each Abelian subgroup (not necessary maximal) of biholomorphisms aC∗-algebra generated by Toeplitz operators. We consider the symplectic manifold (Dn,ω) where ω is the usual symplectic form; using the fact that the action of biholomorphisms is Hamiltonian we give the moment map μH associated with each Abelian subgroup H, where μH:Dn→h∗. Then we introduce the family of symbols S~H which are given by functions of the form a∘μH, where a is an arbitrary bounded function defined on μH(Dn). Finally, we prove that the C∗-algebra generated by Toeplitz operators with symbols in the family S~H is commutative.One important advantage of the point of view of the moment map is that if we consider the symbol set invariant under the action of a nonmaximal Abelian subgroup of biholomorphisms, then theC∗-algebra generated by Toeplitz operators with symbols in this set may not be commutative. In a sense, this symbol set is too big for our purposes, so this method no longer works for nonmaximal Abelian subgroups.The method to find commutativeC∗-algebras of Toeplitz operators using the moment map has not been used in other manifolds and we suppose that we can apply this technique to other symplectic manifolds.


## 2. Weighted Bergman Space and Bergman Projection on the Polydisk

Letn∈N be fixed. Throughout this paper we use the notation z=(z1,…,zn) for a typical element z∈Cn and zj=xj+iyj, j=1,…,n. The unit polydisk is defined by(1)Dn=z∈Cn:zj<1,j=1,…,n.Let -1<λ<+∞; the normalized λ-weighted volume measure dVλ is given by(2)dVλz=λ+1n∏j=1n1-zj2λdAz1⋯dAzn,where z∈Dn and dA(zj)=(1/π)dxjdyj, j=1,…,n.The weighted Bergman spaceAλ2(Dn) is the space of all holomorphic functions which belongs to L2(Dn,dVλ). It is well known (see, e.g., [3]) that Aλ2(Dn) is a closed subspace of L2(Dn,dVλ). The (weighted) Bergman projection BDnλ:L2(Dn,dVλ)→Aλ2(Dn) has the form(3)BDnλfz=∫DnfζKDnλz,ζ¯dVλζ,where the (weighted) Bergman kernel is given by(4)KDnλz,ζ=∏j=1n11-zj¯ζjλ+2.From now on we denote by k=(k1,k2,k3) a vector of nonnegative integer numbers such that n=k1+k2+k3. Denote by Π the upper half-plane; that is,(5)Π=z∈C:Im⁡z>0.Let Mk denote the following domain in Cn:(6)Mk=D×⋯×D×Π×⋯×Π=Dk1×Πk2×Πk3.Consider the Hilbert space L2(Mk,dVλk), where the normalized λ-weighted volume measure has the form(7)dVλkw=λ+1n∏j=1k11-wj2λ∏j=k1+1n2Im⁡wjλdAw1⋯dAwn.Denote by Aλ2(Mk) the (weighted) Bergman subspace of L2(Mk,dVλk). As always, the Bergman space is the subspace of the corresponding L2-space which consists of all analytic functions.The Cayley transformp:Π→D is given by(8)pw=w-iw+i,and the inverse Cayley transform p-1:D→Π acts by the rule(9)p-1z=i1+z1-z.Let id:D→D denote the identity map. The map pk:Mk→Dn given by(10)pkw=idw1,…,idwk1,pwk1+1,…,pwnis obviously a biholomorphism.It is easy to check directly that(11)dVλpw=∏j=k1+1n2iwj+i2λ+2dVλkw,(12)KDnλpkw,pkζ¯=∏j=1k111-wjζ¯·∏j=k1+1nwj+iζ¯j-i2iζ¯j-wjλ+2,(13)1-pζj=2iζj+i,j=k1+1,…,n,where w,ζ∈Mk.The mappk generates the operator(14)Uλ,k:L2Dn,dVλ⟶L2Mk,dVλk,acting by the rule(15)Uλ,kfw=∏j=k1+1np′wjλ/2+1f∘pkw=∏j=k1+1n2iwj+i2λ/2+1f∘pkw.Let f∈L2(Dn,dVλ); changing the variables z=pk(w) and using (11) we have(16)fL2Dn,dVλ2=∫Dnfz2dVλz=∫Mkf∘pkw2∏j=k1+1n2iwj+i2λ+2dVλkw=Uλ,kfL2Mk,dVλk2.The operator Uλ,k is unitary; its inverse operator(17)Uλ,k-1:L2Mk,dVλk⟶L2Dn,dVλhas the form(18)Uλ,k-1gz=∏j=k1+1npk-1′zλ/2+1g∘pk-1z=∏j=k1+1n2i1-zj2λ/2+1g∘pk-1z.It is obvious that Aλ2(Mk)=Uλ,k(Aλ2(Dn)); then the (weighted) Bergman space Aλ2(Mk) is a closed subspace of L2(Mk,dVλk). Changing the variables z=pk(z) and using (11), (12), and (13) we have that the (weighted) Bergman projection Bk,λ of L2(Mk,dVλk) onto Aλ2(Mk) is given by(19)Bk,λgw=Uλ,kBDnλUλ,k-1fw=∫Mk∏j=k1+1n2i21-pζj2wj+i2λ/2+1gζKDnλpkw,pkζ¯dVλpkζ=∫MkgζKMkλw,ζ¯dVλkζ,where the (weighted) Bergman kernel has the form(20)KMkλw,ζ=∏j=1k111-wj¯ζjλ+2·∏j=k1+1niλ+2wj¯-ζjλ+2.


## 3. Representation of the Weighted Bergman Space on the Polydisk

We present a construction of the Bargmann type transformRλ for the case of the polydisk, which maps isometrically the Bergman space onto an appropriate L2 space and which allows us to factor the identity operator in the L2 space and the Bergman projection of the polydisk. This scheme has been already successfully used, for example, in [2, 4–6]; we give here an analogous study for the polydisk.We will use the multi-index notation. For an elementa=(a1,…,am)∈Cm, let |a| denote the sum(21)a=∑j=1maj.Let α∈C denote by [α]m the vector (α,…,α)∈Cm. Introduce in Mk the coordinates(22)zj=zj′=rjtj,rj∈0,1,tj∈S1,j=1,…k1;zk1+j=ζj=xj+iyj,xj∈R,yj∈R+,j=1,…,k2;zk1+k2+j=ζj′=ρjeiθj,ρj∈R+,θj∈0,π,j=1,…,k3.Under the identification(23)z=z1,…,zn=z′,ζ,ζ′=rt,x+iy,ρeiθ=t,x,ρ,r,y,θ,we have(24)Mk=0,1×S1k1×R×R+k2×R×0,πk3=Gk×Ωk,where(25)Gk=Tk1×Rk2×R+k3,Ωk=0,1k1×R+k2×0,πk3.Let (r,y,θ)∈Ωk, we define the measure(26)dηr,y,θ=∏j=1k11-rjλrj∏j=1k22yjλ∏j=1k32λsinλθjdrdydθ.We have(27)dVλkz=dVλkz′,ζ,ζ′=λ+1πnρ1⋯ρk3λ+1ik1t1⋯tk1dtdxdρdηr,y,θ.Then we can write(28)L2Mk,dVλk=L2Tk1⊗L2Rk2⊗L2R+k3⊗L2Ωk,1πnλ+1ndη,where(29)L2Tk1=⨂j=1k1L2S1,dtjitj,L2Rk2=⨂j=1k2L2R,dxj,L2R+k3=⨂j=1k3L2R+,ρjλ+1dρj.Consider the following three unitary operators:(i)
The discrete Fourier transformF:L2(S1,dt/it)→l2=l2(Z) is given by(30)F:f⟼cm=12π∫S1ftt-mdtit;the inverse operatorF-1 has the form(31)F-1:cmm∈Z⟼f=∑m∈Zcmtm.(ii)
The Fourier integral transformF:L2(R)→L2(R) is given by(32)Ffu=12π∫Re-iuxfxdx;the inverse operatorF-1 has the form(33)F-1gx=12π∫Reixugudu.(iii)
The Mellin transformM:L2(R+,ρλ+1)→L2(R) is given by(34)Mψξ=12π∫R+ρ-iξ+λ/2ψρdρ;the inverse operatorM-1 has the form(35)M-1ψρ=12π∫Rρiξ-λ/2-1ψξdξ.Let Zk1=Z×⋯×Z. Introduce the unitary operator(36)Uλ,k=1πnFk1⊗Fk2⊗Mk3⊗I,acting from(37)L2Mk,dVλk=L2Tk1⊗L2Rk2⊗L2R+k3⊗L2Ωk,1πnλ+1ndηonto(38)l2Zk1⊗L2Rk2⊗L2Rk3⊗L2Ωk,λ+1ndη,where F(k1)=F⊗⋯⊗F, F(k2)=F⊗⋯⊗F, M(k3)=M⊗⋯⊗M, and(39)L2Rk3=⨂j=1k3L2R,dρj.In the coordinates z=(rt,x+iy,ρeiθ) the Cauchy-Riemann equations have the form(40)tj2∂∂rj-tjrj∂∂tj,j=1,…,k1,12∂∂xj+i∂∂yj,j=1,…,k2,ρj∂∂ρj+i∂∂θj,j=1,…,k3.Let {ψq}q∈Zk1∈l2(Zk1,L2(Rk2)⊗L2(Rk3)⊗L2(Ωk,(λ+1)ndη)). It straightforward to see that(41)Uλtj2∂∂rj-tjrj∂∂tjUλ-1ψqq∈Zk1mw=12∂ψm-1j∂rj-mj-1ψm-1jrjw,j=1,…,k1,Uλ12∂∂xj+i∂∂yjUλ-1ψqq∈Zk1mw=i2uj+∂∂yjψmw,j=1,…,k2,Uλρj∂∂ρj+i∂∂θjUλ-1ψqq∈Zk1mw=iξj+λ2+1iI+∂∂θjψmw,j=1,…,k3,with w=(u,ξ,r,y,θ)∈Rk2×Rk3×Ωk and m∈Zk1. Here 1j denotes the vector whose jth entry is 1 and zero elsewhere. Thus the image Aλ,k1=Uλ,k(Aλ2(Mk)) consists of all sequences {ψq}q∈Zk1∈l2(Zk1,L2(Rk2)⊗L2(Rk3)⊗L2(Ωk,(λ+1)ndη)) satisfying the equations(42)∂∂rj-mjrjIψm=0,j=1,…,k1,ujI+∂∂yjψm=0,j=1,…,k2,ξj+λ2+1iI+∂∂θjψm=0,j=1,…,k3.These equations are easy to solve; the general solution has the form(43)ψmu,ξ,r,y,θ=cm0u,ξrme-uy-ξ+1+λ/2k3iθ,with cm0∈L2(Rk2)⊗L2(Rk3), m∈Zk1. The function ψm has to be in L2(Rk2)⊗L2(Rk3)⊗L2(Ωk,(λ+1)ndη)), which implies that its support on each variable uj, j=1,…,k2, has to be in R+ and each integer mj, j=1,…,k1, has to be in Z+={0}∪N. That is,(44)ψmu,ξ,r,y,θ=χ+k2uϑmu,ξcmu,ξrme-uy-ξ+1+λ/2k3iθ,where m∈Z+k1=Z+×⋯×Z+, cm∈L2(Rk2)⊗L2(Rk3), χ+k2 is the characteristic function of R+k2=R+×⋯×R+, and (see, e.g., formulas 3.251, 3.381, and 3.892 of [7])(45)ϑmu,ξ=λ+1n∫Ωkr2me-2uy-2ξθdηr,y,θ-1/2=∏j=1k12Γmj+2+λmj!Γ2+λ1/2∏j=1k22ujλ+1λ+1Γλ+11/2∏j=1k3Γλ+2/2+iξjπΓλ+21/2eπξj/2,with uj≥0, j=1,…,k2. Moreover, we have(46)ψmL2Rk2⊗L2Rk3⊗L2Ωk,λ+1ndη2=∫Rk2×Rk3×Ωkχ+k2uϑm2u,ξcmu,ξrme-uy-ξ+1+λ/2k3iθ2λ+1ndudξdηr,y,θ=∫Rk2×Rk3χ+k2uϑm2u,ξcmu,ξ2λ+1n∫Ωkr2me-2uy-2ξθdηr,y,θdudξ=cmL2R+k2⊗L2Rk32.Lemma 1.
The unitary operatorUλ,k=(1/(π)n)F(k1)⊗F(k2)⊗M(k3)⊗I is an isometric isomorphism of the space L2(Mk,dVλk) onto(47)l2Zk1,L2Rk2⊗L2Rk3⊗L2Ωk,λ+1ndη,under which the Bergman space Aλ2(Mk) is mapped onto(48)Aλ,k1=ψm=χ+k2uϑmu,ξcmu,ξrme-uy-ξ+1+λ/2k3iθm∈Z+k1:cm∈L2R+k2⊗L2Rk3.Following [2] we introduce the isometric embedding R0,k acting from the space(49)l2Z+k1,L2R+k2⊗L2Rk3onto(50)l2Zk1,L2Rk2⊗L2Rk3⊗L2Ωk,λ+1ndη,by the rule(51)R0,k:cmu,ξ⟼ψm=χ+k2uϑmu,ξcmu,ξrme-uy-ξ+1+λ/2k3iθ,where cm(u,ξ) is extended by cm(u,ξ)=0, u∈Rk2∖R+k2.The adjoint operatorR0,k∗ acting from(52)l2Zk1,L2Rk2⊗L2Rk3⊗L2Ωk,λ+1ndηonto l2(Z+k1,L2(R+k2)⊗L2(Rk3)) has obviously the form(53)R0,k∗:ψm⟼cm=λ+1n∫Ωkψmu,ξ,r,y,θrme-uy-ξ-1+λ/2k3iθdηr,y,θ.It is easy to check that(54)R0,k∗R0,k=I:l2Z+k1,L2R+k2⊗L2Rk3⟶l2Z+k1,L2R+k2⊗L2Rk3,(55)R0,kR0,k∗=B:l2Zk1,L2Rk2⊗L2Rk3⊗L2Ωk,λ+1ndη⟶Aλ,k1,where B=Uλ,kBMkλUλ,k-1 is the orthogonal projection of(56)l2Zk1,L2Rk2⊗L2Rk3⊗L2Ωk,λ+1ndηonto Aλ,k1.The operatorRλ,k=R0,k∗Uλ,k maps the space L2(Mk,dVλk) onto(57)l2Z+k1,L2R+k2⊗L2Rk3,and the restriction(58)Rλ,kAλ2Mk:Aλ2Mk⟶l2Z+k1,L2R+k2⊗L2Rk3is an isometric isomorphism. The adjoint operator(59)Rλ,k∗=Uλ,k∗R0,k:l2Z+k1,L2R+k2⊗L2Rk3⟶Aλ2Mkis an isometric isomorphism.Remark 2.
We have(60)Rk,λRk,λ∗=I:l2Z+k1,L2R+k2⊗L2Rk3⟶l2Z+k1,L2R+k2⊗L2Rk3,Rk,λ∗Rk,λ=BMkλ:L2Mk,dVλk⟶Aλ2Mk.Theorem 3.
The isometric isomorphismRλ,k∗:l2(Z+k1,L2(R+k2)⊗L2(Rk3))→Aλ2(Mk) is given by(61)Rλ,k∗:cmm∈Z+k1⟼ψz′,ζ,ζ′=12n∑m∈Z+k1∫R+k2×Rk3ϑmu,ξcmu,ξz′meiuζζ′iξ-1+λ/2k3dudξ.Proof.
Let{cm}m∈Z+k1 be a sequence of functions in l2(Z+k1,L2(R+k2)⊗L2(Rk3)); then we have (62)Uλ,k∗R0,kcmm∈Z+k1u,ξ,r,y,θ=12n∑m∈Z+k1∫Rk2×Rk3R0,kcqq∈Z+k1mu,ξtmexuρiξ-1+λ/2k3dudξ=12n∑m∈Z+k1∫R+k2×Rk3ϑmu,ξcmu,ξrtmeiux+iyρeiθiξ-1+λ/2k3dudξ,which proves the theorem.Corollary 4.
The inverse isomorphismRλ,k:Aλ2(Mk)→l2(Z+k1,L2(R+k2)⊗L2(Rk3)) is given by(63)Rλ,k:ψ⟼12nϑmu,ξ∫Mkψz′,ζ,ζ′z′¯me-iuζ¯ζ′¯-iξ-1+λ/2k3dVλkz′,ζ,ζ′m∈Z+k1.


## 4. Toeplitz Operators and Maximal Commutative Subgroups of Biholomorphisms

One of the main results of [8] gives a classification of the maximal Abelian subgroups of biholomorphisms of the unit ball Bn. In particular for the unit disk D or the upper half-plane Π, we have three different types of these subgroups.The group of biholomorphismsAut(D) can be described as follows. Denote by SU(1,1) the special unitary group defined as the set of matrices A of dimension 2×2 such that det⁡A=1 and A∗EA=E, where(64)E=100-1.We have that Aut(D) is realized by the action of SU(1,1) given by(65)SU1,1×D⟶Dabcd·z=az+bcz+d.Definition 5.
LetG be a Lie group. A maximal connected Abelian subgroup H of G is a Lie subgroup of G that satisfies the following:(i)
H is connected Abelian.(ii)
IfH1 is a connected Abelian Lie subgroup of G such that H⊂H1, then H=H1.Definition 6.
LetM and M′ be connected complex manifolds. If H and H′ are subgroups of Aut(M) and Aut(M′), respectively, we will say that (H,M) and (H′,M′) are analytically equivalent if there is a biholomorphism ϕ:M′→M such that H′=φ-1Hφ.It is proved in [8] that every maximal connected Abelian subgroup of Aut(D) is analytically equivalent to one of the following three groups, while neither two from the list are analytically equivalent.The elliptic group of biholomorphisms of the unit disk D is isomorphic to S1 with the following group action:(66)t:z∈D⟼tz∈D,for each t∈S1.The parabolic group of biholomorphisms of the upper half-plane Π is isomorphic to R with the following group action:(67)x:z∈Π⟼x+z∈Π,for each x∈R.The hyperbolic group of biholomorphisms of the upper half-plane Π is isomorphic to R+ with the following group action:(68)ρ:z∈Π⟼ρz∈Π,for each ρ∈R+.Now we turn our attention to the case of the polydisk. We have [3] that Aut(Dn) is realized by the action of SU(1,1)n extended by the nth symmetric group. Consider the Lie group(69)Ωk=Tk1×Rk2×R+k3,acting on Mk=Dk1×Πk2×Πk3 by(70)t,x,ρ:z=z′,ζ,ζ′∈Mk⟼tz′,x+ζ,ρζ′∈Mk,for each (t,x,ρ)∈Ωk. It is evident that this group is analytically equivalent to a maximal Abelian subgroup of SU(1,1)n for each k=(k1,k2,k3). Therefore we have (n+1)(n+2)/2 different types of maximal Abelian subgroups of biholomorphisms of Dn.We have that a functiona(z), z=(rt,x+iy,ρeiθ)∈Mk is invariant with respect to the action of Ωk if and only if a(z)=a(r,y,θ). Hence making use of the results of Section 3 we can give a spectral representation of the Toeplitz operators with symbols invariant under the action of a maximal Abelian subgroup of biholomorphisms.Theorem 7.
Leta=a(r,y,θ) be a bounded measurable function. Then the Toeplitz operator Ta acting on Aλ2(Mk) is unitary equivalent to the multiplication operator γa,kI=Rλ,kTaRλ,k∗. The sequence γa,k={γa,k(m,u,ξ)}m∈Zk1, where u∈R+k2, ξ∈Rk3, is given by(71)γa,km,u,ξ=λ+1nϑm2u,ξ∫Ωkar,y,θr2me-2uy+ξθdηr,y,θ.Proof.
The operatorRλ,k defined in Section 3 factors the identity operator I and the Bergman projection BMkλ (Remark 2); then we have(72)Rλ,kTaRλ,k∗=Rλ,kBMkλaBMkλRλ,k∗=Rλ,kRλ,k∗Rλ,kaRλ,k∗Rλ,kRλ,k∗=R0,k∗Uλ,kar,y,θUλ,k∗R0,k=R0,k∗ar,y,θR0,k.Let {cm(u,ξ)}m∈Z+k1 be a sequence in l2(Z+k1,L2(R+k2)⊗L2(Rk3)). Theorem 3 and Corollary 4 give an integral representation of the operators Rλ,k and Rλ,k∗, respectively; then we have(73)Rλ,kTaRλ,k∗cmu,ξm∈Z+k1=R0,k∗ar,y,θχ+k2uϑmu,ξcmu,ξrme-uy-ξ+1+λ/2k3iθm∈Z+k1=λ+1nϑm2u,ξcmu,ξ∫Ωkar,y,θr2me-2uy+ξθdηr,y,θm∈Z+k1=γa,km,u,ξcmu,ξm∈Z+k1.


## 5. The Moment Map

Let(M,ω) be a symplectic manifold, let G be a Lie group, and let g be the Lie algebra of G. Denote by Sympl(M,ω) the set of all symplectomorphisms of (M,ω). The action(74)ψ:G⟶SMg⟼ψgis called a symplectic action if ψg∈Sympl(M,ω) for all g∈G. In this case we say that G acts by symplectomorphisms.A symplectic actionψ:G→Sympl(M,ω) is a Hamiltonian action if there exists a map(75)μ:M⟶g∗satisfying the following:(1)
For eachX∈g, let(76)μX:M⟶R,p⟼μp,X.LetX♯ be the vector field on M generated by the one-parameter subgroup {etX:t∈R}⊂G; then μX is a Hamiltonian function for the vector field X♯; that is,(77)iX♯ω=dμX.(2)
μ is equivariant with respect to the action ψ and the coadjoint action Ad∗ of G on g∗; that is,(78)μ∘ψg=Ad∗∘μ,for allg∈G.The vector (M,ω,G,μ) is called a Hamiltonian G-space and μ is a moment map.In what follows we will suppose thatG is an Abelian Lie group. It is known that the Lie algebra of a connected Lie group is Abelian if and only if its Lie algebra is Abelian. Therefore if G is Abelian we have that exp⁡(X+Y)=exp⁡Xexp⁡(Y), with X,Y∈g (see [9, Proposition 3.51]). A direct calculation shows the following.Lemma 8.
LetG be a connected Abelian Lie group. If X1,…,Xn∈g, then(79)a1X1+⋯+anXn♯=a1X1♯+⋯+anXn♯,where a1,…,an are real numbers.Moreover, making use of the linearity of the symplectic formω we can state the following result.Proposition 9.
LetG be a connected Abelian Lie group which acts by symplectomorphisms on the symplectic manifold (M,ω). Let X1,…,Xn be a basis of g. If μ:M→g∗ is a map such that μXj is a Hamiltonian function of Xj♯, j=1,…,n, then(80)iX♯ω=dμX,for each X∈g.IfG is Abelian, then the conjugate action of G on G is the trivial action. This implies that the coadjoint action Adg∗ of g∗ on g∗ is the identity map for all g∈G. Consequently, we can restate condition (2) as follows:(2)
μ is invariant with respect to the action ψ; that is,(81)μ∘ψg=μ,for allg∈G.


## 6. The Moment Map of the Polydisk

Let(Dn,ω) be the symplectic manifold where(82)ω=∑j=1n1πdxj∧dyj1-xj2+yj22=∑j=1k1dzj¯∧dzj2πi1-zj22.For each k=(k1,k2,k3) consider the symplectic manifold (Mk,ω′) where (83)ω′=∑j=1k11πdxj∧dyj1-xj2+yj22+∑j=k1+1n1πdxj∧dyj2yj2=∑j=1k1dzj′¯∧dzj′2πi1-zj′22+∑j=1k2dζj¯∧dζj2πi2Im⁡ζj2+∑j=1k3dζj′¯∧dζj′2πi2Im⁡ζ′2.It is easy to check directly that the map(84)pk:w∈Mk⟶w1,…wk1,pwk1+1,…,pwn∈Dn,where p is the Cayley transform defined by 1, is a symplectomorphism of Mk onto the polydisk Dn; that is, ω′=pk∗ω.LetGk denote the Lie group Tk1×Rk2×R+k3. The Lie algebra gk of this group is isomorphic to Rn; then gk∗≅Rn. Let ψ be the action of Gk onto Mk defined by(85)ψt,x,ρ:z=z′,ζ,ζ′∈Mk⟼tz′,x+ζ,ρζ′∈Mk.We note that for each (t,x,ρ)∈Gk the map ψ(t,x,ρ) is a diffeomorphism; moreover we have(86)ψt,x,ρ∗ω′=∑j=1k1∂ψt,x,ρ∂zj′2dzj′¯∧dzj′2πi1-tjzj′22+∑j=1k2∂ψt,x,ρ∂ζj2dζj¯∧dζj2πi2Im⁡xj+ζj2+∑j=1k3∂ψt,x,ρ∂ζj′2dζj′¯∧dζj′2πi2Im⁡ρjζ′2=∑j=1k1tj2dzj′¯∧dzj′2πi1-tjzj′22+∑j=1k2dζj¯∧dζj2πi2Im⁡xj+ζj2+∑j=1k3ρj2dζj′¯∧dζj′2πi2Im⁡ρjζ′2=ω′,which implies that ψ is a symplectic action.We introduce the functionsfj:Mk→R, j=1,…,n, given by(87)fjz=12π1-xj2+yj2,j=1,…,k1,fjz=-14πyj,j=k1+1,…,n-k3,fjz=-xj4πyj,j=n-k3+1,…,n.Lemma 10.
The mapμ:Mk→Rm defined by(88)μz=f1z,…,fmzis a moment map for the symplectic action ψ of Gk into (Mk,ω′); that is, (Mk,ω′,Gk,μ) is a Hamiltonian Gk-space.Proof.
LetX1,…,Xn be the standard basis of gk≅Rn. The vector fields X1♯,…,Xn♯ over Mk generated by the the elements of this basis are given by(89)Xj♯=-yj∂∂xj+xj∂∂yj,j=1,…,k1Xj♯=∂∂xj,j=k1+1,…,k3-nXj♯=xj∂∂xj+yj∂∂yj,j=k3-n+1,…,n.We observe that(90)iXjω′z=1πdxj∧dyj1-xj2+yj22-yj∂∂xjz+xj∂∂yjz,·=-1πyjdyj+xjdxj1-xj2+yj22=dfjz,j=1,…,k1,iXjω′z=1πdx∧dyj2yj2∂∂xjz,·=1πdyj2yj2=dfjz,j=k1+1,…,n-k3,iXjω′z=1πdxsj∧dyj2yj2xj∂∂xjz+yj∂∂yjz,·=1πxjdyj-yjdxj2yj2=dfjz,j=k3-n+1,…,n,with z∈Mk. Hence Lemma 8 implies that(91)iXω′=dμX,for all X∈h. It is easy to check directly that each function fj, j=1,…,n, is invariant under the action of ψ; consequently μ is invariant under this action too. Therefore μ is a moment map.Our aim is to give a moment map for the action of each Abelian subgroup of biholomorphisms of the polydisk. LetH be a Lie subgroup of dimension m≤n of Gk; then the Lie algebra h of H is a Lie subalgebra of gk, which implies that h∗ is a subspace of gk∗. Let π:gk∗→h∗ be the natural projection and consider the symplectic action of H into Mk as the restricted action ψH.Theorem 11.
The mapμH=π∘μ:Mk→h∗ is a moment map for the symplectic action ψH of H into (Mk,ω′); that is, (Mk,ω′,H,μH) is a Hamiltonian H-space.Proof.
LetY1,…,Yk be a basis of h; we can write Yj=a1jX1+⋯+anjXn, j=1,…,k, as a linear combination of the standard basis X1,…,Xn. We observe(92)iYj♯ω=a1jiX1♯ω+⋯+anjiXn♯ω=a1jdf1+⋯+anjdfn=dμHYj.Using Proposition 9 we have that iY♯ω=dμY for every Y in h. Since it is immediate that μH is invariant under the action, we conclude that it is a moment map.


## 7. Symbol Classes Generated by the Moment Map

In this section we introduce a class of bounded measurable functionsS~H over the polydisk associated with each connected Abelian Lie subgroup of biholomorphisms H of Gk=Tk1×Rk2×R+k3 which are a group of isometries over (Mk,ω′,H,μH). Moreover, we have that the C∗-algebra generated by Toeplitz operators with symbols in S~H is commutative.We consider a connected Abelian subgroupH of Gk=Tk1×Rk2×R+k3. In the previous Section we calculate the moment map μH associated with the symplectic action of H; now using this function we define the following set of functions:(93)SH=a:μHMk⊂Rm⟶C.We denote by S~H=μH∗(SH) the set of bounded measurable functions a~:Mk→C such that there exist a∈SH satisfying a~=a∘μH and IH the set of bounded functions over Mk invariant under the restricted action ψH. Moreover, we know that μH is invariant under the action of H, so we have that S~H⊂IH for every H.Definition 12.
We denote byTH the C∗ algebra generated by symbols in S~H where H is an Abelian subgroup of isometries.In particular, we take a maximal Abelian subgroupGk and consider the injective map ϕ:Ωk→Rn defined by (94)r,y,θ⟼12π1-r12,…,12π1-rk12,-14πy1,…,-14πyk2,-14πtan⁡θ1,…,-14πtan⁡θk3,where z=(rt,x+iy,ρeiθ)∈Mk, ϕ(r,y,θ)=μGk(z), and μGk is the moment map associated with Gk. Thus, we obtain that the image ϕ(Ωk)⊂Rn coincides with the image μGk(Mk)⊂Rn. On the other hand, we have that the inverse map ϕ-1:μGk(Mk)→Ωk is given by(95)rj=1-12πuj,j=1,…,k1yj=-14πvj,j=1,…,k2θj=tan-1-14πξj,j=1,…,k3,with (u,v,ξ)∈μGk(Mk)⊂Rn. Hence, we obtain that ϕ∗(IGk)=S~H and (IGk)=(ϕ-1)∗S~H; thus it is clear that IGk=S~H.Therefore, if we consider a maximal Abelian subgroupGk, we have that the C∗-algebra induced by the moment map μGk is equal to the C∗-algebra of Toeplitz operators generated by the set of symbols invariant under the action of Gk. This affirmation is not true for a (nonmaximal) Abelian subgroup of isometries.Example 13.
Let2≤k1=n and we consider the Abelian subgroup H0 as follows: (96)t:z∈Dn⟶tz1,z2,…,zn∈Dn,for each t∈S1. A function a(z), z∈Dn is invariant with respect to this action if and only if a(z)=a(r,z2,…,zn), where z1=reiθ. It is obvious that Toeplitz operators with this class of symbols do not commute.
On the other hand, we observe that a moment map forH0 is given by (97)μH0z=12π1-z12;it follows that an element a∈S~H0 has the form a1/2π1-z12. A straightforward calculus shows that the C∗-algebra TH0 is commutative.Example 14.
LetH1 be the connected Lie subgroup of n-torus Tn which acts as follows: (98)H1×Dn⟶Dn,t,…,t·z⟼tz1,…,tzn.We denote by h1 and tn the Lie algebras corresponding to H1 and Tn, respectively. We know that the Lie algebra h1 is generated by Y=X1+⋯+Xn, where X1,…,Xn is standard basis of tn. Hence the moment map associated with H1 is given by (99)μH1z=12π∑i=1n11-zi2,since we have that (100)iY♯ω=dμH1Y;hence Lemma 10 implies that μH1 is a moment map. We have that every function belonging to S~H1 has the form (101)az=a12π∑i=1n11-zi2.Thus it is clear that the C∗-algebra TH1 is commutative.Remark 15.
ConsiderH0 and H1 as the above examples; its clear that H0∩H1={e}, where e is the identity in the group. In consequence of the above examples we have that TH0∩TH1={Id}, where Id is the operator identity on the Bergman space.In summary, we have that every Abelian groupH of isometries is a subgroup of some maximal Abelian group Gk. Hence, we have that S~H⊂S~Gk=IGk. From this we obtain the following theorem.Theorem 16.
IfH is a connected Abelian Lie group of biholomorphisms in Dn, then the C∗-algebra TH is commutative.Remark 17.
ConsiderH1 and H2 Abelian Lie groups of biholomorphisms in Dn such that H1 is subgroup of H2; under the considerations of the above theorem we have that TH1 is subalgebra of TH2. Therefore a chain of commutative Abelian groups of isometries induce a chain of commutative C∗-algebras.



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