TOEPLITZ OPERATORS WITH QUASI-RADIAL QUASI ...

arXiv:1201.2161v1 [math.OA] 10 Jan 2012

TOEPLITZ OPERATORS WITH QUASI-RADIAL QUASI-HOMOGENEOUS SYMBOLS AND BUNDLES OF LAGRANGIAN FRAMES RAUL QUIROGA-BARRANCO and ARMANDO SANCHEZ-NUNGARAY Abstract. We prove that the quasi-homogenous symbols on the projective space Pn (C) yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball Bn . These algebras are Banach but not C ∗ . We prove the existence of a strong link between such symbols and algebras with the geometry of Pn (C). The latter is also proved for the corresponding symbols and algebras on Bn .

1. Introduction The study of commutative algebras of Toeplitz operators has shown to be a very interesting subject. Some previous results in this topic serve as background to this work. First, it was shown the existence of symbols defining interesting commutative C ∗ -algebras of Toeplitz operators on bounded symmetric domains (see [2], [7], [8] and [9]). Also, it was exhibited in [11] the existence of Banach algebras, which are not C ∗ , of commutative Toeplitz operators on the unit ball Bn . And, in [6] we constructed commutative C ∗ -algebras of Toeplitz operators on complex projective spaces. A remarkable fact is that the currently known commutative C ∗ -algebras of Toeplitz operators on Bn are naturally associated to Abelian subgroups of the group of biholomorphisms of Bn . In fact, their systematic description is best understood with the use of such groups of biholomorphisms (see [8] and [9]). Furthermore, this provided the guiding light to construct commutative C ∗ -algebras of Toeplitz operators in the complex projective space Pn (C): the currently known C ∗ -algebras for Pn (C) are naturally associated and described from the maximal tori of the group of isometric biholomorphisms of Pn (C) (see [6]). The known Banach (not C ∗ ) algebras of commutative Toeplitz operators first introduced in [11] for Bn are given by the so called quasi-homogeneous symbols. Such symbols are defined in terms of radial and spherical coordinates of components in Bn (see Section 3 below). However, their introduction lacked the stronger connection with the geometry of the domain observed for the commutative C ∗ -algebras of Toeplitz operators on Bn . Given these lines of research, there are two natural problems to consider. First, to determine whether or not there are any interesting Banach algebras, that are not 1991 Mathematics Subject Classification. Primary 47B35; Secondary 32A36, 32M15, 53C12. Key words and phrases. Toeplitz operators, commutative Banach algebras, Lagrangian frames, complex projective space. The first named author was partially supported by SNI-Mexico and by a Conacyt grant. The second named author was partially supported by a Conacyt postdoctoral fellowship. 1

2

RAUL QUIROGA-BARRANCO and ARMANDO SANCHEZ-NUNGARAY

C ∗ , of commutative Toeplitz operators on Pn (C). And second, to find any possible special links between the geometry of Bn and the Banach algebras, introduced in [11], of commutative Toeplitz operators. The goal of this work is to solve these problems. On one hand, we define quasi-homogeneous symbols in the complex projective space Pn (C), and show that these provide Banach algebras of commutative Toeplitz operators on every weighted Bergman space of Pn (C). The results that exhibit the commuting Toeplitz operators are obtained in Section 4, where Theorem 4.5 is the main result. On the other hand, we also prove that the Banach algebras defined by quasi-homogeneous symbols turn out to have a strong connection with the geometry of the supporting space, for both Bn and Pn (C); the main results in this case are presented in Section 5. In particular, we prove that the quasi-homogeneous symbols, for both Bn and Pn (C), can be associated to an Abelian group of holomorphic isometries of the corresponding space (see Theorem 5.1). Such group is a subgroup of a maximal torus in the corresponding isometry group. We further prove that the groups associated to quasi-homogeneous symbols afford pairs of foliations with distinguished Lagrangian and Riemannian geometry known as Lagrangian frames (see Section 5 below and [7], [8] and [9]). This recovers the behavior observed for the C ∗ -algebras of commutative Toeplitz operators constructed in [8], [9] and [6], for which such Lagrangian frames appear as well. Nevertheless, it is important to note a key difference between the C ∗ case and the Banach case. For the C ∗ -algebras of Toeplitz operators on Bn and Pn (C), as constructed in [8], [9] and [6], the Lagrangian frames are obtained for the whole space, i.e. they come from Lagrangian submanifolds of the whole space, either Bn or Pn (C). But for the Banach algebras defined by quasi-homogeneous symbols the Lagrangian frames are obtained on the fibers of a suitable fibration over either Bn or Pn (C). The existence of such fibration and the Lagrangian frames on its fibers are obtained in Theorem 5.4. It is also proved in Theorem 5.6 that a full maximal torus of isometries continues to play an importante role, since the complement to the group that defines the fiberwise Lagrangian frames acts by automorphisms of such frames. The authors want to acknowledge the input of Pedro Luis del Angel, from Cimat, provided through several conversations with him.

2. Preliminaries on the Geometry and Analysis of Pn (C) In this section we establish our notation concerning the n-dimensional complex projective space Pn (C). For further details we refer to the bibliography. There is a natural realization of Cn as an open conull dense subset given by Cn → Pn (C),

z 7→ [1, z],

which defines a biholomorphism onto its image. Note that the points of Pn (C) are denoted by [w], the complex line through w ∈ Cn+1 \ {0}. We will refer to this embedding as the canonical embedding of Cn into Pn (C). Let us denote with ω the canonical Kähler structure on Pn (C) that defines the Fubini-Study metric, whose volume is then given by Ω = (ω/2π)n . These induce

QUASI-RADIAL QUASI-HOMOGENEOUS SYMBOLS AND LAGRANGIAN FRAMES

3

on Cn the following K¨ ahler form and volume element, respectively Pn Pn (1 + |z|2 ) k=1 dzk ∧ dz k − k,l=1 z k zl dzk ∧ dz l ω0 = i , (1 + |z|2 )2 dV (z) 1 Ω0 = n , π (1 + |z1 |2 + · · · + |zn |2 )n+1 where dV (z) denotes the Lebesgue measure on Cn . Let H denote the dual bundle of the tautological line bundle of Pn (C). We recall that H carries a canonical Hermitian metric h obtained from the (flat) Hermitian metric of Cn+1 . Then, it is also well known that the curvature Θ of (H, h) satisfies the identity Θ = −iω, which amounts to say that (H, h) is a quantum line bundle over Pn (C). We will denote with Γ(Pn (C), H m ) and Γhol (Pn (C), H m ) the smooth and holomorphic sections of H m , respectively. Note that H m denotes the m-th tensorial power of H. Clearly, both of these spaces lie inside L2 (Pn (C), H m ). For every m ∈ Z+ and with respect to the canonical embedding of Cn into Pn (C), we define the weigthed measure on Pn (C) with weight m by Ω(z) (n + m)! m! (1 + |z1 |2 + · · · + |zn |2 )m (n + m)! dV (z) = . π n m! (1 + |z1 |2 + · · · + |zn |2 )n+m+1

dνm (z) =

A simply computation shows that dνm is a probability measure for all m ∈ Z+ . For simplicity, we will use the same symbol dνm to denote the weighted measures for both Pn (C) and Cn . It is also straightforward to show that the canonical embedding of Cn into Pn (C) induces a canonical isometry Φ : L2 (Cn , νm ) → L2 (Pn (C), H m ) with respect to which we will identify these spaces in the rest of this work. Also, we will denote with h·, ·im the inner product of this Hilbert spaces. The weighted Bergman space on Pn (C) with weight m ∈ Z+ is defined by: A2m (Pn (C)) = {ζ ∈ L2 (Pn (C), H m ) : ζ is holomorphic} = Γhol (Pn (C), H m ). These Bergman spaces are finite-dimensional and are described by the following well known result. Proposition 2.1. For every m ∈ Z+ , the Bergman space A2m (Pn (C)) satisfies the following properties. (i) A2m (Pn (C)) can be identified with the space P (m) (Cn+1 ) of homogeneous polynomials of degree m over Cn+1 . (ii) For Φ : L2 (Cn , νm ) → L2 (Pn (C), H m ) the canonical isometry described above, we have Φ(A2m (Pn (C))) = Pm (Cn ), the space of polynomials on Cn of degree at most m. In what follows, we will use this realization of the Bergman spaces without further notice.

4


Recall the following notation for multi-indices α, β ∈ Nn and z ∈ Cn |α| = α1 + · · · + αn , α! = α1 ! . . . αn !,

α ∈ Nn ,

z α = z α1 . . . z αn , δα,β = δα1 ,β1 . . . δαn ,βn . A2m (Pn (C)) n

The Bergman space has a basis consisting of the polynomials z α = αn . . . zn where α ∈ N and |α| ≤ m. Hence we will consider the set

z1α1

Jn (m) = {α ∈ Nn : |α| ≤ m}. More precisely, an easy computation shows that the set ( ) 21 m! α (2.1) z : α ∈ Jn (m) α!(m − |α|)! is an orthonormal basis of A2m (Pn (C)). For ψ ∈ L2 (Pn (C), H m ), and considering the identification Φ, we define the Bergman projection by Z ψ(w)K(z, w)dV (w) (n + m)! Bm (ψ)(z) = n+m+1 π n m! (1 + w n 1 w 1 + · · · + wn w n ) C where

K(z, w) = (1 + z1 w1 · · · + zn wn )m . This operator satisfies the well known reproducing property. Proposition 2.2. If ψ ∈ L2 (Pn (C), H m ), then Bm (ψ) belongs to the weighted Bergman space A2m (Pn (C)). Also, Bm (ψ) = ψ if ψ ∈ A2m (Pn (C)). Using this, we define the Toeplitz operator Ta on A2m (Pn (C)) with bounded symbol a ∈ L∞ (Pn (C)) by Ta (ϕ) = Bm (aϕ), for every ϕ ∈ A2m (Pn (C)). Let us denote by Sn the unit sphere in Cn . In this work, we will use the following identity on the n-sphere Sn Z 2π n α! β (2.2) ξ α ξ dS(ξ) = δα,β = (n − 1 + |α|)! Sn where dS be the corresponding surface measure on Sn (see [12]).

3. Toeplitz operators with quasi-homogeneous symbols Quasi-homogeneous symbols were introduced in [11] on the unit ball Bn in Cn . However, we will also consider quasi-homogeneous symbols for the projective space Pn (C) as functions defined on Cn . The basic definitions of the quasi-radial and quasi-homogeneous symbols are very similar for both Bn and Cn , so we will recall them here together. This will also be useful latter on for our common geometric treatment of quasi-homogeneous symbols on Bn and Pn (C). To simplify our notation, from now on Un will denote one of either Bn or Cn , corresponding to whether we are dealing with Bn or Pn (C), respectively. For the latter case, we are considering Cn canonically embedded into Pn (C) as described in Section 2.


5

Let k = (k1 , . . . , kl ) ∈ Zl+ be a multi-index so that |k| = n. We will call such multi-index k a partition of n. For the sake of definiteness, we will assume that k1 ≤ · · · ≤ kl . This partition provides a decomposition of the coordinates z ∈ Un as z = (z(1) , . . . , z(l) ) where z(j) = (zk1 +···+kj−1 +1 , . . . , zk1 +···+kj ), for every j = 1, . . . , l, and the empty sum is 0 by convention. For z ∈ Un , we define rj = |z(j) | and z(j) ξ(j) = rj if z(j) 6= 0. Besides the quasi-radii (r1 , . . . , rl ), this provides a set of coordinates (ξ(1) , . . . , ξ(l) ) ∈ Sk1 × · · · × Skl . For a partition k of n, a k-quasi-radial symbol a on Un is a function a : Un → C which depends only on the coordinates (r1 , . . . , rl ) introduced above. In other words, a(z) = a(r1 , . . . , rl ). The set of all k-quasi-radial functions is denoted by Rk . The family of sets Rk , while k varies over the partitions of n, is a partially ordered by inclusion. The minimal element among these sets is the set R(n) of radial functions and the maximal element is the set R(1,...,1) of separately radial functions. Also, for k a partition of n, a k-quasi-homogeneous symbol ϕ on Un is a function ϕ : Un → C of the form q ϕ(z) = a(r1 , . . . , rl )ξ p ξ where a is a k-quasi-radial symbol and p, q ∈ Nn satisfy p · q = p1 q1 + · · · + pn qn = 0. Recall that every k the family Rk is contained in R(1,...,1) . Hence, the Toeplitz operators Ta for symbols a ∈ Rk can be simultaneously diagonalized with respect to the monomial basis in the corresponding Bergman space (see [8] and [6]). Furthermore, for the case Un = Cn the following result provides the multiplication operator so obtained. Note that this result is similar to Lemma 3.1 from [11]. Lemma 3.1. Consider the case Un = Cn , and let k be a partition of n. For any k-quasi-radial bounded measurable symbol a(r1 , . . . , rl ), we have Ta z α = γa,k,m (α)z α , for every α ∈ Jn (m), where γa,k,m (α) = γa,k,m (|α(1) |, . . . , |α(l) |) (3.1)

2l (n + m)! Ql (m − |α|)! j=1 (kj − 1 + |α(j) |) Z l Y 2|α |+2kj −1 rj (j) drj a(r1 , . . . , rl )(1 + r2 )−(n+m+1) × =

Rn +

j=1

Proof. Let α ∈ Zn+ with |α| ≤ m. Then, we have hTa z α , z α im = haz α , z α im Z a(r1 , . . . , rl )z α z α dV (z) (n + m)! . = 2 2 n+m+1 π n m! Cn (1 + |z1 | + · · · + |zn | )

6


Making the substitution z(j) = rj ξ(j) , where rj ∈ [0, ∞) and ξ(j) ∈ Skj , for j = 1, . . . , l, we obtain

haz α , z α im =

×

Z

(n + m)! π n m! l Z Y

j=1

Skj

a(r1 , . . . , rl )(1 + r2 )−(n+m+1)

Rn +

α

l Y

2|α(j) |+2kj −1

rj

drj

j=1 α(j)

ξ(j)(j) ξ (j) dS(ξ(j) )

2m α!(n + m)! Ql m! j=1 (kj − 1 + |α(j) |) Z l Y 2|α |+2kj −1 rj (j) drj a(r1 , . . . , rl )(1 + r2 )−(n+m+1) × =

Rn +

j=1

and the result follows from (2.2)

We now find the action of the Toeplitz operators with quasi-homogeneous symbols on the canonical monomial basis. Note again that the following result corresponds to Lemma 3.3 from [11]. q

q

Lemma 3.2. Consider the case Un = Cn , and let aξ p ξ = a(r1 , . . . , rl )ξ p ξ be a k-quasi-homogeneous symbol for k a partition of n. Then, the Toeplitz operator Taξp ξq acts on monomials z α with α ∈ Zn+ and |α| ≤ m as follows

α

Taξp ξq z =

(

γã,k,p,q,m (α)z α+p−q 0

for α + p − q ∈ Jn (m) for α + p − q 6∈ Jn (m)

where 2l (α + p)!(n + m)! Ql (α + p − q)!(m − |α + p − q|)! j=1 (kj − 1 + |α(j) + p(j) |) Z l Y 2|α +p −q |+2kj −1 rj (j) (j) (j) drj a(r1 , . . . , rl )(1 + r2 )−(n+m+1) ×

γã,k,p,q,m (α) =

(3.2)

Rn +

j=1

Proof. Let α, β ∈ Zn+ satisfy |α|, |β| ≤ m. Then, we have q

hTaξp ξ q z α , z β im = haξ p ξ z α , z β im Z q a(r1 , . . . , rl )ξ p ξ z α z β dV (z) (n + m)! . = 2 2 n+m+1 π n m! Cn (1 + |z1 | + · · · + |zn | )


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Applying the change of the variables z(j) = rj ξ(j) , where rj ∈ [0, ∞) and ξ(j) ∈ Skj , for j = 1, . . . , l, this yields

q

haξ p ξ z α , z β im = ×

(n + m)! π n m! l Y

Z

a(r1 , . . . , rl )(1 + r2 )−(n+m+1)

Rn +

|α(j) |+|β(j) |+2kj −1

rj

drj

j=1

×

l Z Y

Skj

j=1

(3.3)

= δα+p,β+q ×

Z

α

ξ(j)(j)

m!

+p(j) β(j) +q(j) dS(ξ(j) ) ξ (j)

Ql

2l (α + p)!(n + m)!

j=1 (kj

− 1 + |α(j) + p(j) |)

a(r1 , . . . , rl )(1 + r2 )−(n+m+1)

Rn +

×

l Y

2|α(j) +p(j) −q(j) |+2kj −1

rj

drj

j=1

Observe that this expression is non zero if and only if β = α + p − q, which a priori belongs to Jn (m). We conclude the result from the orthonormality of the basis defined in (2.1).

4. Commutativity results for quasi-homogeneous symbols on Pn (C) In the rest of this section we will restrict ourselves to the case Un = Cn . The results in this section show that the commuting identities proved in [11] for the unit ball Bn have corresponding ones for the complex projective space Pn (C). Theorem 4.1. Let k ∈ Zl+ be a partition of n and p, q ∈ Nn a pair of orthogonal multi-indices. If a1 and a2 are non identically zero k-quasi-radial bounded symbols, then the Toeplitz operators Ta1 and Ta2 ξp ξq commute on each weighted Bergman space A2m (Pn (C)) if and only if |p(j) | = |q(j) | for each j = 1, . . . , l. Proof. Let α ∈ Jn (m) be given. First note that if α + p − q 6∈ Jn (m), then the Lemmas 3.1 and 3.2 imply that both Ta1 Ta2 ξp ξq z α and Ta2 ξp ξq Ta1 z α vanish. Hence, we can assume that α + p − q ∈ Jn (m). Applying again Lemmas 3.1 and 3.2 we

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obtain 2m (α + p)!(n + m)! Ql (α + p − q)!(m − |α + p − q|)! j=1 (kj − 1 + |α(j) + p(j) |) Z l Y 2|α +p −q |+2kj −1 2 −(n+m+1) rj (j) (j) (j) drj a2 (r1 , . . . , rl )(1 + r ) ×

T a1 T a2 ξ p ξ q z α =

Rn +

j=1

l

2 (n + m)! Ql (m − |α + p − q|)! j=1 (kj − 1 + |α(j) + p(j) − q(j) |) Z l Y 2|α +p −q |+2kj −1 rj (j) (j) (j) drj a1 (r1 , . . . , rl )(1 + r2 )−(n+m+1) ×

×

Rn +

j=1

× z α+p−q . And similarly, we have

2l (n + m)! Ql (m − |α|)! j=1 (kj − 1 + |α(j) |) Z l Y 2|α |+2kj −1 2 −(n+m+1) rj (j) drj a1 (r1 , . . . , rl )(1 + r ) ×

T a2 ξ p ξ q T a1 z α =

Rn +

j=1

l

2 (α + p)!(n + m)! Ql (α + p − q)!(m − |α + p − q|)! j=1 (kj − 1 + |α(j) + p(j) |) Z l Y 2|α +p −q |+2kj −1 rj (j) (j) (j) drj a2 (r1 , . . . , rl )(1 + r2 )−(n+m+1) ×

×

Rn +

j=1

× z α+p−q

From which we conclude that Ta1 Ta2 ξp ξ q z α = Ta2 ξp ξq Ta1 z α if and only if |p(j) | = |q(j) | where j = 1, . . . , l. If we assume that |p(j) | = |q(j) | for all j = 1, . . . , l, then the equations (3.1) and (3.2) combine together to yield the following identity. 2l (α + p)!(n + m)! Ql (α + p − q)!(m − |α + p − q|)! j=1 (kj − 1 + |α(j) + p(j) |) Z l Y 2|α |+2kj −1 rj (j) drj a(r1 , . . . , rl )(1 + r2 )−(n+m+1) ×

γã,k,p,q,m (α) =

Rn +

j=1

(α + p)! j=1 (kj − 1 + |α(j) |) γa,k,m (α) Q (α + p − q)! lj=1 (kj − 1 + |α(j) + p(j) |) l Y (α(j) + p(j) )!(kj − 1 + |α(j) |) γa,k,m (α) = (α(j) + p(j) − q(j) )!(kj − 1 + |α(j) + p(j) |) j=1

=

(4.1)

Ql

As a consequence of the previous computations, we also obtain the following very special property of Toeplitz operators with quasi-homogeneous symbols.


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Corollary 4.2. Let k ∈ Zl+ be a partition of n and p, q ∈ Nn a pair of orthogonal multi-indices such that |p(j) | = |q(j) | for all j = 1, . . . , l. Then for each non identically zero k-quasi-radial function a, we have Ta Tξp ξq = Tξp ξ q Ta = Taξp ξq . Consider k = (k1 , . . . , kl ) and a pair of multi-indices p, q such that p ⊥ q and |p(j) | = |q(j) | for j = 1, . . . l. we define p˜(j) = (0, . . . , 0, p(j) , 0, . . . , 0),

q˜(j) = (0, . . . , 0, q(j) , 0, . . . , 0)

where the only possibly non zero part is placed in the j-th position. In particular, we have p = p˜(1) + . . . + p˜(l) and q = q˜(1) + . . . + q˜(l) . Now let Tj = Tξp˜(j) ξp˜(j) for every j = 1, . . . , l. As a consequence of the previous computations we obtain the following result. Corollary 4.3. The Toeplitz operators Tj = Tξp˜(j) ξp˜(j) , for j = 1, . . . l mutually commute and m Y Tj = Tξ p ξ q j=1

Next, we obtain a necessary and sufficient condition for two given quasi-homogeneous symbols to determine Toeplitz operators that commute with each other.

Theorem 4.4. Let k ∈ Zl+ be a partition of n and p, q ∈ Nn a pair of orthogq v onal multi-indices. Consider a(r1 , . . . , rl )ξ p ξ and b(r1 , . . . , rl )ξ u ξ two k-quasin homogeneous symbols on P (C) where a(r1 , . . . , rl ) and b(r1 , . . . , rl ) are k-quasiradial measurable and bounded symbols. Assume that |p(j) | = |q(j) | and |u(j) | = |v(j) | for all j = 1, . . . , l. Then, the Toeplitz operators Taξp ξq and Tbξu ξv commute on each weighted Bergman space A2m (Pn (C)) if and only if for each j = 1, . . . , l one of the following conditions hold (i) pj = qj = 0 (ii) uj = vj = 0 (iii) pj = uj = 0 (iv) qj = vj = 0 Proof. First, we observe that for our hypotheses, the quantities Tbξu ξv Taξp ξq z α and Taξp ξq Tbξu ξv z α are always simultaneously zero or non zero. Hence, we compute Taξp ξq Tbξu ξv z α and Tbξu ξ v Taξp ξq z α , for α ∈ Jn (m), assuming that both are non zero. By (4.1), we have the following expression 2l (α + p)!(n + m)! Q (α + p − q)!(m − |α|)! lj=1 (kj − 1 + |α(j) + p(j) |) Z l Y 2|α |+2kj −1 rj (j) drj a(r1 , . . . , rl )(1 + r2 )−(n+m+1) ×

Tbξu ξv Taξp ξ q z α =

Rn +

j=1

l

2 (α + p − q + u)!(n + m)! Ql (α + p − q + u − v)!(m − |α|)! j=1 (kj − 1 + |α(j) + u(j) |) Z l Y 2|α |+2kj −1 b(r1 , . . . , rl )(1 + r2 )−(n+m+1) × rj (j) drj

×

Rn +

×z

α+p−q+u−v

j=1

.

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Similarly, we also have 2l (α + u)!(n + m)! Ql (α + u − v)!(m − |α|)! j=1 (kj − 1 + |α(j) + u(j) |) Z l Y 2|α |+2kj −1 rj (j) drj b(r1 , . . . , rl )(1 + r2 )−(n+m+1) ×

Taξp ξq Tbξu ξv z α =

Rn +

j=1

2l (α + u − v + p)!(n + m)! × Q (α + p − q + u − v)!(m − |α|)! lj=1 (kj − 1 + |α(j) + p(j) |) Z l Y 2|α |+2kj −1 rj (j) drj a(r1 , . . . , rl )(1 + r2 )−(n+m+1) × Rn +

×z

j=1

α+p−q+u−v

Therefore, we conclude that Tbξu ξv Taξp ξq z α = Taξp ξq Tbξu ξ v z α if and only if (αj + pj − qj + uj )!(αj + pj ) (αj + uj − vj + pj )!(αj + uj )! = (αj + uj − vj )! (αj + pj − qj )! for j = 1, . . . , l. And this equality holds if and only if for each j = 1, . . . , l one of the following conditions is fulfilled (i) pj = qj = 0, (ii) uj = vj = 0, (iii) pj = uj = 0, (iv) qj = vj = 0. Finally, we present one of our main results: the construction of a Banach algebra of Toeplitz operators on Pn (C). Our construction is parallel to the one presented in [11]. As above, continue to consider k ∈ Zl+ a partition of n, and now let h ∈ Zl+ be such that 1 ≤ hj ≤ kj − 1, for all j = 1, . . . , l. Let ϕ be a k-quasi-homogeneous symbol on Cn of the form q

ϕ(z) = a(r1 , . . . , rl )ξ p ξ , where p, q ∈ Nn satisfy p · q = 0. Consider the decompositions of p and q given as follows p = (p(1) , . . . , p(l) ), p(j) = (pj,1 , . . . , pj,kj ),

q = (q(1) , . . . , q(l) ) q(j) = (qj,1 , . . . , qj,kj ),

for j = 1, . . . , l. With respect to this decomposition, we will now assume that pj,r = 0,

qj,s = 0,

for r > hj and s ≤ hj , where j = 1, . . . , l. As before, we also assume that |p(j) | = |q(j) | for all j, which now becomes pj,1 + · · · + pj,hj = qj,hj +1 + · · · + qj,kj . Let us denote with Rk (h) the space of all symbols obtained through this construction. As a consequence of Theorem 4.4.


11

Theorem 4.5 (Banach algebra of symbols with commuting operators). For k and h as above, the Banach algebra of Toeplitz operators generated by the symbols in Rk (h) is commutative on each weighted Bergman space on Pn (C). 5. Bundles of Lagrangian frames and quasi-homogeneous symbols In the rest of this work, Rk (h) will denote the space of symbols described above for Un ; recall that the latter is either Cn or Bn . Whenever necessary, we will specify which of these two choices is under consideration. Recall that the n-torus Tn acts by isometric biholomorphisms on Un by the expression t · z = (t1 z1 , . . . , tn zn ) where t ∈ Tn and z ∈ Un . The following result associates to the space of symbols Rk (h) a subgroup of the Abelian group of isometries corresponding to the Tn -action. Theorem 5.1 (Torus associated to Rk (h)). Let us consider the space of symbols Rk (h) on Un where k ∈ Zl+ is a partition of n, and consider the subgroup of Tn given by l Y T1[kj ] Tk = j=1

kj

where 1[kj ] denotes the element of C all of whose coordinates are 1. Then, for the Tn -action by isometries on Un , Tk is precisely the subgroup of Tn that leaves invariant the elements of Rk (h). In other words, for t ∈ Tn we have t ∈ Tk ⇐⇒ ϕ(tz) = ϕ(z) for every ϕ ∈ Rk (h), z ∈ Un . Proof. Associated to the partition k = (k1 , . . . , kl ), we recall the decomposition z = (z(1) , . . . , z(l) ) introduced before and note that we have a corresponding decomposition t = (t(1) , . . . , t(l) ) for every element t ∈ Tn . With this notation at hand, we observe that for t ∈ Tn the following two conditions are equivalent • t ∈ Tk , • there exist s ∈ Tl such that (5.1)

t(j) = sj 1[kj ] for all j = 1, . . . , l.

First, let us consider ϕ ∈ Rk (h) and t ∈ Tk as given in equation (5.1). If q ϕ(z) = a(r)ξ p ξ , then q

ϕ(tz) = a(r)(tξ)p (tξ)q = a(r)tp t ξ p ξ = ϕ(z)

l Y

j=1

= ϕ(z),

p

q

(j) (j) t(j) t(j) = ϕ(z)

l Y

q

|p(j) | |q(j) | sj

sj

j=1

where we have used that |p(j) | = |q(j) | for all j = 1, . . . , l, since ϕ ∈ Rk (h); note that we have also used that the quasi-radial symbols a(r) is Tn -invariant. Conversely, let us assume that t ∈ Tn satisfies ϕ(tz) = ϕ(z), for every ϕ ∈ Rk (h) and z ∈ U∗n . We will pick a particular choice of p, q ∈ Nn . Given 1 ≤ j0 ≤ l choose

12


r, s such that 1 ≤ r ≤ hj < s ≤ kj and define ( 1 if j = j0 and i = r, (p(j) )i = 0 otherwise, ( 1 if j = j0 and i = s, (q(j) )i = 0 otherwise. q

Then, it is easy to check that the symbol ϕ(z) = ξ p ξ belongs to Rk (h). For this symbol, we have ϕ(tz) = (t(j0 ) )r (t(j0 ) )s ϕ(z), and so the invariance of ϕ under t implies that (t(j0 ) )r = (t(j0 ) )s for all r, s satisfying 1 ≤ r ≤ hj < s ≤ kj for our arbitrarily given 1 ≤ j0 ≤ l. This shows that t satisfies the conditions given by equation (5.1) and thus completes the proof. The set of symbols Rk (h) is determined by the partition k ∈ Zl+ of n and h ∈ Zl+ (satisfying the properties described before). We will in turn associate to k a fibration whose fibers carry distinguished symplectic geometry and that can be naturally associated to the set of symbols Rk (h). Let us start by defining the map πk : Pn (C) →

l Y

j=1

[z0 , z] 7→

Pkj −1 (C) 1 1 z(1) , . . . , z(l) , z0 z0

where z0 ∈ C and z ∈ Cn . Then, πk is a rational map defined in the complement of the algebraic variety Pk = {[z0 , z] ∈ Pn (C) : z0 ∈ C \ {0}, z(j) ∈ Ckj \ {0}, j = 1, . . . , l}. In particular, it is well defined and holomorphic in the subset Vk =

l Y

(Ckj \ {0}) ⊂ Cn ⊂ Pn (C),

j=1

where the last inclusion is given by the canonical embedding z 7→ [1, z] introduced before. In fact, with this realization of Vk as subset of Pn (C) we have (5.2)

πk (z) = ([z(1) ], . . . , [z(l) ]),

for every z ∈ Vk . We also note that the set Vk is the natural domain of definition of the symbols that belong to Rk (h). We now consider the complexification of the torus Tk introduced above. More precisely, we denote l Y C∗ 1[kj ] , Ak = j=1

where, as in the statement of Theorem 5.1, 1[kj ] denotes the element of Ckj all of whose coordinates are 1. Note that Ak is a subgroup of C∗n isomorphic to C∗l by


13

an obvious isomorphism with the same expression as the one described above for Tk and Tl . Recall, that the group C∗n acts on Pn (C) biholomorphically through the assignment (a, [z0 , z]) 7→ [z0 , a1 z1 , . . . , an zn ], ∗n for a ∈ C , z0 ∈ C and z ∈ Cn . Furthermore, this action restricts to the natural componentwise action of C∗n on Cn . In particular, the restriction of this action to Ak is given on Vk as follows. Ak × Vk → Vk (5.3)

((a1 1[k1 ] , . . . , al 1[kl ] ), z) 7→ (a1 z(1) , . . . , al z(l) ),

Also, note that the restriction of such action to Tk ⊂ Ak yields the action that leaves invariant the symbols of Rk (h). With this respect, the following result relates the set of symbols Rk (h) to the rational map πk , while providing geometric information on the latter. We note that our notion of principal fiber bundle is as defined in [4]. Theorem 5.2 (Principal bundle associated to Rk (h)). Let k ∈ Zl+ be a partition of n. Then, the rational map πk satisfies the following property. Q • The Ak -action on Vk turns the map πk |Vk : Vk → lj=1 Pkj −1 (C) into a complex principal fiber bundle with structure group Ak . Ql In particular, πk |Vk : Vk → j=1 Pkj −1 (C) is a holomorphic submersion. Also, for Ql every p ∈ j=1 Pkj −1 (C), the fiber πk−1 (p) is a holomorphic l-dimensional submanifold of Vk ⊂ Cn . Proof. Using the expressions for πk given in (5.2) we note that the πk |Vk is the product of the canonical quotient maps Ckj \ {0} → Pkj −1 (C),

j = 1, . . . , l,

which are well known to be principal fiber bundles with structure group C∗ . Then, the expression for the Ak -action given by (5.3) implies that πk is indeed a principal fiber bundle with structure group Ak . The other claims follow from elementary properties of fiber bundles and submersions. We recall the definition of Lagrangian frame introduced in [8, 9, 7]. We refer to these references for further details on the notions involved. Definition 5.3. On a K¨ ahler manifold N , a Lagrangian frame is a pair (O, F) of smooth foliations that satisfy the following properties. • Both foliations are Lagrangian. In other words, the leaves of both foliations are Lagrangian submanifolds of N . • If L1 and L2 are leaves of O and F, respectively, then Tx L1 ⊥ Tx L2 at every x ∈ L1 ∩ L2 . • The foliation O is Riemannian. I.e. the Riemannian metric of N is invariant by the leaf holonomy of O. • The foliation F is totally geodesic. I.e. its leaves are totally geodesic submanifolds of N . We will refer to O and F as the Riemannian and totally geodesic foliations, respectively, of the Lagrangian frame.

14


The next result shows that the fibers of the submersion πk carry Lagrangian frames naturally associated to the symbols Rk (h). Note that the result holds for both the projective geometry and hyperbolic geometry of Pn (C) and Bn , respectively, even though their K¨ ahler structures are different. We recall that, with our previous convention, Un denotes either Cn ⊂ Pn (C) or Bn . And we also emphasize that Un carries either the complex projective or complex hyperbolic Kähler metric according to the choice of either case. Theorem 5.4 (Lagrangian frames associated to Rk (h)). For a partition k ∈ Zl+ of n, let Rk (h) be the set of quasi-radial quasi-homogeneous symbols on Un . Then, Ql every fiber of the holomorphic submersion πk |Vk ∩Un : Vk ∩ Un → j=1 Pkj −1 (C) carries a Lagrangian frame for the K¨ ahler structure inherited from Un so that the symbols belonging to Rk (h) are constant on the leaves of the Riemannian foliation of the frame. More precisely, the following conditions hold for every fiber Fp = πk−1 (p) ∩ Vk ∩ Un , where p ∈ πk (Vk ∩ Un ). (i) The action of Tk restricted to Fp defines a Riemannian foliation Op on whose leaves every symbol that belongs to Rk (h) is constant. (ii) The vector bundle T Op⊥ defined as the orthogonal complement of T Op inside of T Fp is integrable to a totally geodesic foliation J Op . (iii) The pair (Op , J O p ) is a Lagrangian frame of the complex manifold Fp for the K¨ ahler structure inherited from Un . Proof. We fix a fiber Fp as described in the statement. We will also denote Fbp = πk−1 (p), so that Fp ⊂ Fbp with proper inclusion precisely in the case Un = Bn . Note that both Fp and Fbp are complex l-dimensional submanifolds of Cn . By Theorem 5.2, the map πk is Ak -invariant, and since Un is Tk -invariant, it follows that Fp is Tk -invariant. Also, the action of Ak on Fbp is free and so Tk acts freely on Fp thus defining a foliation Op with real l-dimensional leaves. We recall that the Tk -action is a restriction of the natural Tn -action on Cn . Since the latter is isometric on Un , then the Tk -action is isometric as well. This last property implies that the foliation Op is Riemannian (see, for example, [7]). Moreover, by Theorem 5.1 the symbols that belong to Rk (h) are Tk -invariant and so constant on the leaves of Op . This proves (1). It is known that the Tn -action on Un defines a Lagrangian foliation: the Tn orbits are Lagrangian. This has been verified in [7] and [6] for Un = Bn and Un = Cn ⊂ Pn (C), respectively. Since Tk is a subgroup of Tn , we conclude that ω(Tz Op , Tz Op ) = 0 for every z ∈ Fp and for the Kähler form ω of Un . Also, we observe that the real dimension of the leaves of Op and the complex dimension of the fiber Fp have the common value l. This implies that Op is a Lagrangian foliation of Fp . Since the orthogonal complement of a Riemannian foliation is totally geodesic (see, for example, [7]), to prove (2) and (3) it suffices to show that T Op⊥ = iT Op is integrable. To prove the integrability of iT Op let us consider the vector field on Un given by the expression (Xj )z = (iδj1 z(1) , . . . , iδjl z(l) ),

j = 1, . . . , l,

where δjr denotes Kronecker’s delta for j, r = 1, . . . , l. Then, Xj is the vector field on Un induced by the action of a one-parameter subgroup of Tk . More precisely,


15

for the action ψj : R × Un → Un (θ, z) 7→ (eiδj1 θ z(1) , . . . , eiδjl θ z(l) ) we have

d ψj (θ, z). dθ θ=0 Note that either a direct computation or the fact that Tk is Abelian implies that (Xj )z =

[Xj1 , Xj2 ] = 0,

(5.4)

for all j1 , j2 = 1, . . . , l. Also, we observe that (X1 )z , . . . , (Xl )z defines a basis for Tz Op for every z ∈ Fp . A similar construction provides an explicit expression for the local flow associated to the vector field JXj , for j = 1, . . . , l. Here and in what follows, J denotes the complex structure of Un . We now consider the assignment βj : (r, z) 7→ (rδj1 z(1) , . . . , rδjl z(l) ),

j = 1, . . . , l,

for which we now have d βj (r, z) = (δj1 z(1) , . . . , δjl z(l) ) = (JXj )z . dr r=1 We note that the local flow defined by βj is holomorphic. We observe that (JX1 )z , . . . , (JXl )z is a basis for iTz Op for every z ∈ Fp ; this is a consequence of the corresponding property for Tz Op stated above. Next, we observe that, for j1 , j2 = 1, . . . , l, we have [JXj1 , JXj2 ] = J[Xj1 , JXj2 ] = J 2 [Xj1 , Xj2 ] = 0. Here we used in the first and second identities the fact that JXj2 and Xj1 , respectively, define Lie derivatives that commute with J; the latter is a consequence of the fact that both fields have holomorphic local flows (see [5]). For the third identity, we used the equations (5.4). Thus, we have proved that the bundle iT Op has a set of sections that generate the fibers and commute pairwise. Hence, the integrability of iT Op follows from Frobenius’ Theorem. Finally, we prove that a suitable complement of Ak in C∗n acts by symmetries of the bundle obtained in Theorem 5.2. For a partition k ∈ Zl+ of n, let us consider the following subgroups. For every j = 1, . . . , l we denote Bj = {z ∈ (C∗ )kj : z1 · · · zkj = 1} and consider the group given by Bk =

l Y

Bj ⊂ C∗n .

j=1

Lemma 5.5. For any partition k ∈ Zl+ of n the map Ak × Bk → C∗n (a, b) 7→ ab

16


is an isomorphism of Lie groups. Proof. For a given integer m ∈ Z+ consider the group given by B = {z ∈ C∗m : z1 · · · · · zm = 1}. It is immediate that the product map C∗ × B → C∗m (z, b) 7→ (zb1 , . . . , zbm ), is an isomorphism of Lie groups. The statement of the Lemma follows from this fact and the definitions of Ak and Bk . The group Bk has a natural action on Pn (C) as a subgroup of C∗n . Also, for every j = 1, . . . , l the group Bj , as a subgroup of C∗kj , has a natural action on Pkj −1 (C). By taking the product of these actions, we obtain an action of Bk on Ql kj −1 (C). Furthermore, it is easy to verify that for these actions we have j=1 P πk (b · z) = bπk (z),

for every b ∈ Bk and z ∈ Vk . This implies the first part of the following result. The second part brings into play the subgroup of Tn that can be considered the “complement” of Tk . We refer to [4] for the definition of the automorphism of a principal bundle. Ql Theorem 5.6. The Bk -actions on Vk and j=1 Pkj −1 (C) yield principal bundle Ql automorphisms πk |Vk : Vk → j=1 Pkj −1 (C). In particular, the action of Tn ∩ Bk is an isometric action on Vk ∩ Un that permutes the fibers of πk |Vk ∩Un and their Lagrangian frames defined in Theorem 5.4. Proof. As noted above, only the second part requires justification. First note that Tn acts isometrically and so Tn ∩Bk acts isometrically as well. Also, the Lagrangian frames of the fibers are given by Tk -orbits and their orthogonal complements. But Tn ∩ Bk preserves the former since the Bk -action commutes with the Tk -action. Finally, Tn ∩ Bk preserves the orthogonal complement of the Tk -orbits because the action of Tn ∩ Bk is isometric. References [1] P. Griffiths, J. Harris, Principles of algebraic geometry. Reprint of the 1978 original. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. [2] S. Grudsky, R. Quiroga-Barranco, N. Vasilevski, Commutative C -algebras of Toeplitz operators and quantization on the unit disk, J. Funct. Anal. 234(2006), no. 1, 1–44. [3] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80. Academic Press, Inc., New York-London, 1978. [4] S. Kobayashi, K. Nomizu, Foundations of differential geometry. Vol. I. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. [5] S. Kobayashi, K. Nomizu, Foundations of differential geometry. Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. [6] R. Quiroga-Barranco, A. Sanchez-Nungaray, Commutative C ∗ -algebras of Toeplitz operators on the complex projective space, Integral Equations and Operator Theory, 71(2011), no. 2, 225–243 [7] R. Quiroga-Barranco, N. Vasilevski, Commutative algebras of Toeplitz operators on the Reinhardt domains, Integral Equations Operator Theory, 59(2007), no. 1, 67–98.


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[8] R. Quiroga-Barranco, N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators on the unit ball. I. Bargmann-type transforms and spectral representations of Toeplitz operators, Integral Equations Operator Theory, 59(2007), no. 3, 379–419. [9] R. Quiroga-Barranco, N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators on the unit ball. II. Geometry of the level sets of symbols, Integral Equations Operator Theory, 60(2008), no. 1, 89–132. [10] M. Schlichenmaier, Berezin-Toeplitz quantization for compact Kahler Manifolds. A review of results, Adv. Math. Phys., 2010, Art. ID 927280, 38 pp. [11] N. Vasilevski, Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators, Integral Equations Operator Theory, 66(2010), no. 1, 141–152. [12] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer Verlag, 2005. ´ n en Matema ´ ticas, Guanajuato, M´ Raul Quiroga-Barranco, Centro de Investigacio exico E-mail address: [email protected] ´ n en Matema ´ ticas, Guanajuato, Armando Sanchez-Nungaray, Centro de Investigacio M´ exico E-mail address: [email protected]