Toeplitz Operators, Pseudo-Homogeneous Symbols, and Moment Maps on the Complex Projective Space
Following previous works for the unit ball due to Nikolai Vasilevski, we define quasi-radial pseudo-homogeneous symbols on the projective space and obtain the corresponding commutativity results for Toeplitz operators. A geometric interpretation of these symbols in terms of moment maps is developed. This leads us to the introduction of a new family of symbols, extended pseudo-homogeneous, that provide larger commutative Banach algebras generated by Toeplitz operators. This family of symbols provides new commutative Banach algebras generated by Toeplitz operators on the unit ball.
The study of commutative algebras generated by Toeplitz operators has been a very interesting subject in the last decade. The first works studied commutative -algebras generated by Toeplitz operators on the unit disk and the unit ball, where an important connection with the geometry of the underlying space was found (see [1–3]). In particular, the commutativity of the -algebras generated by Toeplitz operator with symbols invariant under the action of maximal Abelian groups was proved. A dual study was carried out for the complex projective space in . It was proved that results similar to those of the unit ball hold for projective spaces, but in this case the symbols that yield commutative -algebras are those that depend only on the radial part of the homogeneous coordinates.
On the other hand, Vasilevski  introduced a new family of commutative algebras generated by Toeplitz operators which are Banach algebras but not -algebras. These Banach algebras are those generated by Toeplitz operators whose symbols are the so-called quasi-radial quasi-homogeneous symbols on the unit ball. This was again brought into the setup of the projective space in . In this latter work, it was proved that the quasi-radial quasi-homogeneous symbols yield commutative Banach algebras generated by Toeplitz operators on each weighted Bergman space of the complex projective space. A remarkable fact found in  is that a geometric description of the symbols was also given by proving that the quasi-homogeneous symbols can be associated with an Abelian subgroup of holomorphic isometries and that one can also construct Lagrangian foliations over principal bundles over such Abelian subgroups. Furthermore, this was proved for both the projective space and the unit ball.
On the other hand, García and Vasilevski considered in  new families of commutative Banach algebras generated by Toeplitz operators over the Bergman space of the two-dimensional unit ball. These are obtained by a generalization of the quasi-homogeneous symbols over the two-dimensional unit ball.
Recently Vasilevski  introduced two further and more general commutative Banach algebras over the unit ball. These algebras are generated by Toeplitz operator where the symbols are called quasi-radial pseudo-homogeneous symbols, with a variation to consider a product of spheres or a single sphere. Moreover, Vasilevski showed that the quasi-radial quasi-homogeneous symbols of both types have interesting properties. More precisely, if and are symbols from the product of spheres type, then . However, when the symbols and are from the case of a single sphere, then one still has but .
Our goal is to extend the existence of such type of commutative Banach algebras generated by Toeplitz operators from the unit ball to the setup of the complex projective space. Thus we introduce the quasi-radial pseudo-homogeneous symbols for projective spaces, both the multisphere and single-sphere cases. Furthermore, we prove that the pseudo-homogeneous symbols have a geometric background which is supplied by the moment maps associated with Hamiltonian actions. The latter are given in our case by toral actions on projective spaces that are restricted to the usual toral actions on complex spaces. In particular, our geometric interpretation for the pseudo-homogeneous symbols applies to the case of the unit ball as well. This allows us to raise some question about the existence of larger families of symbols that properly contain the pseudo-homogeneous ones. This motivates us to consider what we called extended pseudo-homogeneous symbols that embrace all the possibilities discussed so far for both the unit ball and the projective space.
The paper is organized as follows. After some preliminary remarks presented in Section 2, in Sections 3 and 4 we prove the existence of projective versions (dual to the unit ball) of commutative Banach algebras generated by Toeplitz operators whose symbols are quasi-radial pseudo-homogeneous, first for the multisphere case and then for the single-sphere case. This is achieved on each weighted Bergman space. In Section 5 we consider some aspects of symplectic geometry as Hamiltonian actions, moment maps, symplectic reductions, and Delzant polytopes. Using these tools, we give a geometric description of quasi-radial pseudo-homogeneous symbols and we indicate how these techniques may be used to extend these families of symbols. Thus, we introduce in Section 6 the extended pseudo-homogeneous symbols, which contain many families contained up to this point. In particular, we exhibit commutative Banach algebras generated by Toeplitz operator with extended quasi-radial pseudo-homogeneous symbols. It is important to note that for this kind of symbols we have that in general. Finally, we show in Section 7 how to obtain commutative Banach algebras in the case of the unit ball for our extended pseudo-homogeneous symbols.
2. Geometry and Analysis on
On the -dimensional complex projective space and for every we consider the open setwhere the natural homogeneous coordinates are given by the holomorphic chart defined byThese yield the holomorphic atlas of .
The canonical Kähler structure on is given by the closed -formon for every , whereThe corresponding Riemannian metric is the well-known Fubini-Study metric. The volume element of with respect to the Fubini-Study metric is defined by
On the other hand, the hyperplane line bundle on is given bywhich assigns to every point in the dual of the line in that such point represents. The bundle has a natural Hermitian metric inherited from the usual Hermitian inner product on . As usual, let us denote by the th tensor power of and by its corresponding Hermitian inner product. Hence, we denote by the -completion of the space of smooth sections with respect to the Hermitian inner product:where . Since is compact, the space of global holomorphic sections of , denoted by , is finite dimensional and so closed in . This yields the so-called weighted Bergman space on with weight .
We note that there is a trivialization of on given by This map together with yields an isometry , whereand denotes the Lebesgue measure on (see  for further details). With respect to this isometry, the Bergman space is identified with the space of polynomials of degree at most on which is denoted by .
In the rest of this work we will use the multi-index notation without further notice. In particular, the set of monomials on of degree at most is a basis for . This basis is indexed by the set . And it is easy to compute thatis an orthonormal basis of .
On the other hand, the Bergman projection for is given byFinally, the Toeplitz operator on with bounded symbol is given byfor every .
3. Quasi-Radial Pseudo-Homogeneous Symbols
Let be a multi-index so that . This partition provides a decomposition of the coordinates of as , wherefor every , and the empty sum is by convention. We further decompose as follows. For every we define . And for any such we writewhereCorrespondingly, we write and . We observe that for every we have , the subset of the real -sphere with nonnegative coordinates.
Definition 1. A symbol is called -pseudo-homogeneous if it has the formwhere and with .
A particularly useful case is that where with , .
It is easily seen from the definition of the variables involved that the value of a pseudo-homogeneous symbol is well defined in terms of homogeneous coordinates. In fact, we observe that the condition is used to have a well-defined expression in Definition 1. This can be checked, for example, following the arguments from Section in . Furthermore, the proof of Lemma from the previous reference ensures that we can take without any loss of generality. Hence, from now on we will consider partitions of the form where is a partition of and we will also call the corresponding symbols -pseudo-homogeneous. With such restriction, the general form of a pseudo-homogeneous symbol is given bysince .
This convention allows us to consider a natural corresponding expression for the -pseudo-homogeneous symbols on through the canonical embedding given by the chart . More precisely, the -pseudo-homogeneous symbol associated with and on is given by
Next, we consider the quasi-radial symbols on as first introduced in .
Definition 2. Let be a partition of . A -quasi-radial symbol is a function that satisfiesfor some function which is homogeneous of degree 0.
A similar discussion as the one provided above applies in this case (see ). Hence, we will consider and -quasi-radial symbols on for . Their corresponding expression is given bydefined for and some function.
Collecting the previous two types of symbols we obtain the following.
Definition 3. Let be a partition of . A -quasi-radial -pseudo-homogeneous symbol is a function of the form where is -quasi-radial symbol and is a -pseudo-homogeneous symbol.
By the previous discussion, for a partition of we will consider symbols defined on of the formwhere satisfies .
It follows from Lemma in  that for a given -quasi-radial symbol the Toeplitz operator acting on satisfiesfor every , where is given by
We now extend to quasi-radial pseudo-homogeneous symbols the previous result and the corresponding one for quasi-homogeneous symbols found in . Note that the following computation is dual to the one found in . In what follows we will denote . We will also consider the following sets:
Theorem 4. Let be a -quasi-radial -pseudo-homogeneous symbol whose expression is given as in (21). Then, the Toeplitz operator acting on the weighted Bergman space satisfiesfor every , where
Proof. It is enough to compute the following inner product in the Bergman space .The last product of integrals is nonzero if and only if , and in this case it equals . Hence, we will assume that for which we haveIf for every we denote by the volume element on , apply spherical coordinates on , and use the identity , then we now haveApplying the identity together with the corresponding expression of in terms of the coordinates on we haveOn the other hand, we havebecause we assumed that . The result now follows from this.
The following is a particular case.
Corollary 5. For as above, one assumes that for every . Then, the function from Theorem 4 satisfies the following:for every such that , and it is zero otherwise.
We observe that Corollary 5 reduces to formula (23) for quasi-radial symbols. Furthermore, for a symbol of the formwhere is fixed and satisfies and for every , Corollary 5 gives the following expression:In particular, the Toeplitz operator with this latter symbol is independent of the weight .
As a consequence of the previous discussion we have the following result.
Corollary 6. Let be a symbol as in Corollary 5. Then, one hasFurthermore, the Toeplitz operators , , for , pairwise commute andIn particular, the Toeplitz operators for these symbols generate a commutative Banach algebra.
4. Single-Sphere Pseudo-Homogeneous Symbols
For a given we consider the decompositionwhere and , for every . We also introduce the coordinatesfor every , where . Hence, we have . And with this notation we can write
Definition 7. For a partition of , a symbol is called single-sphere -pseudo-homogeneous if it has the formwhere and are computed for , the function , and satisfies for every .
We observe that with this definition the part of the symbols is well defined since it is given in terms of homogeneous coordinates. And as before the condition on ensures that the part depending on is well defined as well; this has to be considered since the symbol is a function of the class and not just the corresponding homogeneous coordinate . On the other hand, note that is given on the complement of a hyperplane of and so it is well defined as an function.
The difference between the single-sphere -pseudo-homogeneous symbols and the -pseudo-homogeneous symbols from Definition 1 is that for the latter we partition the coordinates into several spheres according to the partition . In particular, we can refer to the symbols from Definition 1 as multisphere -pseudo-homogeneous symbols.
As before, a particularly interesting case is the one for which we havewhere , where is the subspace of corresponding to the coordinates . For simplicity, we will denote by this latter set. Also note that if then we have for every that essentially by definition.
We now fix a partition as above and we will now consider a symbol of the formwhere . As before, we can assume our symbol defined for . Also note that for the multi-index needs to be given only on the coordinates in the set , but it still has to satisfy . Since we need to consider multi-indexes for all the coordinates of , we introduce the following notation. Given , we will denote by the multi-index such thatIn other words, vanishes outside the coordinates in and coincides with in these coordinates.
Theorem 8. Let be a single-sphere -pseudo-homogeneous symbol whose expression is given as in (42). Then, the Toeplitz operator acting on the weighted Bergman space satisfiesfor every , where
Proof. As before we proceed to calculate as follows: The last product of integrals is nonzero precisely when , in which case its value is . So we now assume that We now replace for every and apply spherical coordinates in to obtain the following where is the volume element of :Now we parametrize by coordinates as before. To simplify our computations we assume that so that We now apply the change of coordinates for and for Since we can writenote that the action of the operator does not depend on the parameter . From the previous results we also obtain the following.
Corollary 9. For a partition of , let be a -quasi-radial symbol and for every let be a symbol of form (42). Then the Toeplitz operators , pairwise commute on every weighted Bergman space .
Remark 10. It is easy to check that for suitably chosen symbols as above. This should be compared with Corollary 6.
5. Geometric Description of Symbols through Moment Maps
We will consider a compact group acting on a manifold , which in our case will be compact as well. For simplicity and without loss of generality we will assume that the -action is free: the stabilizer of every point is trivial. We also note that any action of a compact Lie group is necessarily proper.
For every we will denote by the vector field on whose flow is . More precisely, we haveWe further assume that the manifold carries a symplectic form and that the -action preserves . In particular, the -form given byis closed for every . The symplectic -action on is called Hamiltonian if there exists a smooth mapwhich is -equivariant for the coadjoint -action on (where denotes the dual vector space of the Lie algebra ) and if we havefor every . Here, the function is defined by The map is called the moment map of the Hamiltonian -action on .
Marsden-Weinstein theorem allows performing what is known as the symplectic point reduction of a Hamiltonian action. More precisely, for the above setup, if is a regular point of the moment map , then the quotientadmits a manifold structure such that the quotient mapdefines a principal bundle with structure group . Furthermore, there exists a unique symplectic form on that satisfies The symplectic manifold is called a symplectic point reduction of .
This provides a well-known alternative definition of the projective space that we now describe (see  for further details). On we consider the canonical symplectic form and the symplectic action This -action is Hamiltonian with moment map We recall that is the Lie algebra of , and since the latter is Abelian the adjoint and coadjoint actions are trivial. The symplectic point reduction in this case reduces to the well-known fact thatIt is also well known that the corresponding symplectic form is precisely the one introduced in Section 2.
Our second fundamental example is given by the actionwhere we have identifiedA straightforward computation shows that for the natural embedding used before the subset is -invariant with restricted action given bywhere is now realized as the usual product of circles . This -action is Hamiltonian. In fact, if we realize the Lie algebra of asthen the moment map of the -action on is given by Note that is an -dimensional affine space, but it carries a natural structure of vector space by choosing the pointas its origin.
Definition 11. A toric manifold is a compact symplectic manifold with dimension together with an effective Hamiltonian -action.
A fundamental result due to Delzant establishes a one-to-one correspondence between toric manifolds and a certain family of polytopes.
Theorem 12 (Delzant). If is -dimensional toric manifold with moment map , then the image is a polytope that satisfies the following properties.
Simple. There are edges meeting at each vertex.
Rational. The edges meeting at a vertex are of the form for , where .
Smooth. The vectors can be chosen to be a basis for .
A convex polytope satisfying these properties is called a Delzant polytope. Furthermore, there is a one-to-one correspondence
Delzant’s theorem applied to the above -action on yields the following Delzant polytope given by the following expression and their corresponding realizations: and the latter is precisely the set introduced in (24). As a consequence, we obtain the following remark.
Remark 13. The expressions found in Sections 3 and 4 that describe the action on monomials of the Toeplitz operators with pseudo-homogeneous symbols (both single and multisphere) are all given as integrals on the Delzant polytopes of projective spaces. Furthermore, the same remark applies to the corresponding results for the unit ball: Lemmas and from . These facts allow us to raise the following questions. (i)Is there a relationship between the quasi-radial pseudo-homogeneous symbols and Delzant polytopes?(ii)With the introduction of the notion of Delzant polytopes, is it possible to infer the existence of further symbols whose associated Toeplitz operators can be explicitly described on monomials?
We now proceed to answer the first question in Remark 13. Hence, we fix a partition of as before.
5.1. Geometric Description of -Quasi-Radial Symbols
Let us consider the action (64) restricted to the torus As before we have made the identificationSince the previous action is the restriction of a Hamiltonian action it is Hamiltonian as well. Furthermore, the corresponding moment map is given by This follows from the fact that the -action is a restriction of the action given by (64), for which the moment map is already known, and the results in Section are from .
The following result describes the -quasi-radial symbols as the pullback under the moment map of bounded functions on the polytope of such map.
Theorem 14. A bounded symbol is -quasi-radial if and only if it belongs to , where , the polytope associated with the moment map . More precisely, is -quasi-radial if and only if there exists such that .
Proof. Suppose that for some . Then, we can writewhich is clearly the composition of the map with a homogeneous function of degree . Hence, is -quasi-radial by Definition 2.
Conversely, suppose that is a bounded -quasi-radial symbol. By definition we can writefor some function which is homogeneous of degree . In particular, we haveand then we see that belongs to by a trivial change of coordinates.
5.2. Geometric Description of Symbols of the Form
Besides the partition we now fix such that for every . In particular, and according to the notation from Section 3, the symbol is -pseudo-homogeneous.
Corresponding to the -action on given by (72) we have an induced quotient map
On the other hand, the condition for every implies that the symbol is -invariant. As a consequence we now have the following.
Theorem 15. Every symbol of the type belongs to . More precisely, one hasIn other words, any such symbol can be considered as a function on the product of projective spaces .
5.3. Geometric Description of the Pseudo-Homogeneous Symbols of the Form
We will now consider a bounded function and the corresponding -pseudo-homogeneous symbol wherefor all