# Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space ofℂn

Authors: Jorge Luis Arroyo Neri; Armando Sánchez-Nungaray; Mauricio Hernández Marroquin; Raquiel R. López-Martínez
Journal: Journal of Function Spaces
 (2021)

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## Abstract

We introduce the so-called extended Lagrangian symbols, and we prove that theC∗-algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space ℂn is isomorphic and isometric to the C∗-algebra of matrix-valued functions on a certain compactification of ℝn obtained by adding a sphere at the infinity; moreover, the matrix values at the infinity points are equal to some scalar multiples of the identity matrix.


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## Body

## 1. Introduction

Letm∈ℕ, the one-dimensional m poly-Fock space Fm2ℂ⊂L2ℂ,dμ consists of all m-analytic functions φ which satisfy
(1)∂∂z¯mφ=12m∂∂x+i∂∂ymφ=0,where dμ=π−1e−z·z¯dxdy is the Gaussian measure in ℂ and dxdy is the Euclidian measure in ℝ2=ℂ. Further, the one-dimensional true poly-Fock space of order m is given by
(2)Fm2ℂ=Fm2ℂ⊝Fm−12ℂ.In the case of several variables, forn∈ℕ, the n-dimensional Gaussian measure in ℂn is given by dμnz=π−ne−z2dxdy, where dxdy is the Euclidian measure in ℝ2n. We have that the space L2ℂn,dμn is the tensorial product of n components
(3)L2ℂn,dμn=L2ℂ,dμ⊗⋯⊗L2ℂ,dμ,and the Fock space F2ℂn is
(4)F2ℂn=F2ℂ⊗⋯⊗F2ℂ.Given a multi-indexα=α1,⋯,αn∈ℤ+n, the poly-Fock space Fα2ℂn of order α is given by
(5)Fα2ℂn=⊗j=1nFαj2ℂ.Similarly, the true poly-Fock spaceFα2ℂn is
(6)Fα2ℂn=⊗j=1nFαj2ℂ.In [1], Vasilevski introduced the poly-Fock spaces over ℂn and he obtained the following decomposition formula:
(7)L2ℂn,dμn=⊕α=n∞Fα2ℂn.Moreover, he showed that the true poly-Fock spaceFα2ℂn is isomorphic and isometric to L2ℝn,dx⊗H~α−1, where H~α−1 is the one-dimensional space generated by the function
(8)h~α−1y=∏j=1nhαj−1yj,and each hαj−1yj is a Hermite’s function in ℝ.Another treatment of the poly-Fock spaces can be found in [2], where the author characterized all lattice sampling and interpolation sequences in the poly-Fock spaces. He introduced the polyanalytic Bargmann transform from vector-valued Hilbert spaces to poly-Fock spaces, and he showed the duality between sampling and interpolation in polyanalytic spaces and multiple interpolation and sampling in analytic spaces.The Toeplitz operators acting on the Fock space have been investigated by several authors. For example, in [3], the authors studied Toeplitz operators acting on the one-dimensional Fock space and on true poly-Fock space whose symbols are bounded radial functions that have a finite limit at the infinity. They considered an orthonormal basis of normalized complex Hermite polynomials to prove that the radial operators are diagonal. In [4], the authors studied Toeplitz operators acting on the one-dimensional poly-Fock space with horizontal symbols such that the limit values at x=∞ and x=−∞ exist. They proved that the C∗-algebra generated with this class of symbols is isomorphic to the C∗-algebra of functions on ℝ¯ with values on the m×m matrices, whose limit value at x=∞ and x=−∞ are equal to some scalar multiples of the identity matrix. In [5], the authors introduced the Toeplitz operators with L-invariant symbols over the Fock space F2ℂn for a Lagrangian plane L, and they proved that the corresponding C∗-algebra generated is isometric to the C∗-algebra generated by Toeplitz operators with horizontal symbols.On the other hand, the spaces of homogeneously polyanalytic functions have been studied recently. For example, in [6], the authors computed the reproducing kernel of the Bergman space of homogeneously polyanalytic functions on the unit ball in ℂn and on the Siegel domain.The main result of this paper is the following: theC∗-algebra generated by Toeplitz operators with extended Lagrangian symbols acting on the homogeneously poly-Fock space over ℂn is isomorphic and isometric to the C∗-algebra of matrix-valued functions on a certain compactification of ℝn with the sphere at the infinity; moreover, the values at the infinity points are scalar multiplies of the identity matrix.This paper is organized as follows. In Section2, we define the so-called homogeneously poly-Fock space and study some of its properties. In Section 3, we prove that every Toeplitz operator with a horizontal symbol acting on the poly-Fock (or homogeneously poly-Fock) space is unitary equivalent to a multiplication operator by a matrix-valued function. In Section 4, we introduce the concept of extended horizontal symbol and we describe the C∗-algebra generated by Toeplitz operators with this kind of symbols acting on both the poly-Fock space and the homogeneously poly-Fock space. Finally, in Section 5, we define the extended Lagrangian symbols and we prove that the C∗-algebra generate by Toeplitz operators with these symbols acting on the homogeneously poly-Fock space is isomorphic to the C∗-algebra generated by Toeplitz operators with horizontal symbols acting on the same space.


## 2. Poly-Fock Spaces overℂn

In this section, we define the homogeneously poly-Fock space and we review some facts about the classic poly-Fock spaces.Forz=z1,⋯,zn∈ℂn, we denote x=x1,⋯,xn, y=y1,⋯,yn∈ℝn, where zj=xj+iyj. The Fock space F2ℂn is given by
(9)F2ℂn=F2ℂ⊗⋯⊗F2ℂ.We have thatF2ℂn is a Hilbert space with the usual inner product of functions:
(10)f,g=1πn∫ℂnfzgz¯e−z2dxdy.Moreover, the Fock spaceF2ℂn is a reproducing kernel Hilbert Space, whose reproducing kernel is given by
(11)Kw,z=Kwz=ez·w¯.For the multi-indexα=α1,⋯,αn∈ℤ+n, we recall the operations α=α1+⋯+αn, α!=α1!⋯αn!, and zα=z1α1⋯znαn, with z∈ℂn. Also, for α,β∈ℤ+n, we have α±β=α1±β1,⋯,αn±βn. We consider the set ℤ+n with the lexicographic order. Finally, for α=α1,⋯,αn∈ℤ+n, we write Pα=α1⋯αn. For the multi-index 1n=1,⋯,1, sometimes, we write 1 instead of 1n as long as no there is no confusion.Letα∈ℤ+n be a multi-index and consider the poly-Fock space Fα2ℂn defined in (5), since every one-dimensional poly-Fock space Fαj2ℂ is a direct sum of true poly-Fock spaces whose order is less than or equal to αj, see [1], p. 5-6, we have
(12)Fα2ℂn=⊕λ1=1α1Fλ12ℂ⊗⋯⊗⊕λn=1αnFλn2ℂ=⊕λi≤αiλ∈ℤ+nFλ2ℂn.Note that the number of components in (12) is equal to Pα=α1⋯αn.Now, letk∈ℕ be a natural number such that k≥n.Definition 1.
The homogeneously poly-Fock space of orderk over ℂn is given by
(13)Fk2ℂn=⊕α=kFα2ℂn.
The number of multi-indices whose absolute value is exactlyk is equal to sk=k−1n−1.Definition 2.
The poly-Fock space of orderk in ℂn is given by
(14)Fk2ℂn=⊕∣α∣=nkFα2ℂn.
The number of multi-indices whose absolute value is less than or equal tok is equal to sk=kk−n.Remark 3 (see [6], Proposition 2.7).
The authors introduced the concept of homogeneously polyanalytic function; this concept is very important in the development of this paper. Also, they proved that homogeneously polyanalytic spaces are invariant under linear change of variables.In [1], Vasilevski applied the “creation” and “annihilation” operators in the Fock spaces and he proved the following results:
(1)
All true poly-Fock spaces are isomorphic one to each other(2)
The explicit expression of the functionsψz in the true poly-Fock space Fα2ℂn is given by(15)ψz=∑λ∈ℤ+nλi≤αi−1λα−1!λ!α−1−λ!z¯α−1−λ∂λφz,where φz∈F2ℂn and ∂λφ=∂λφ/∂z1λ1⋯∂znλn.(3)
The reproducing kernel of the true poly-Fock spaceFα2ℂn can be obtained applying the “creation” operator to the reproducing kernel of the Fock space F2ℂnRemark 4.
Using the creation operator defined in [1], we have that the homogeneously poly-Fock space Fk2ℂn and the poly-Fock space Fk2ℂn are reproducing kernel Hilbert spaces.On the other hand, forα=α1,⋯,αn∈ℤ+n and y=y1,⋯,yn∈ℝn, consider the function introduced in (8):
(16)h~α−1y=∏j=1nhαj−1yj,where α−1=α1−1,⋯,αn−1 and each hαj−1yj is Hermite’s function in ℝ. The set h~α−1y, with α=n,⋯,∞, forms an ortonormal base in the space L2ℝn,dy. We denote H~α−1⊂L2ℝn,dy to the one-dimensional space generated by the function h~α−1y, whose orthogonal projection is given by
(17)P~α−1ψy=h~α−1y∫ℝnψηh~α−1ηdη.Now, we consider the operators(18)U1:L2ℂn,dμn⟶L2ℝ2n,dxdy,U1φx,y=π−n/2e−x+iyx−iy/2φx+iy.U2=I⊗F:L2ℝn,dx⊗L2ℝn,dy⟶L2ℝn,dx⊗L2ℝn,dy,with Fψy=2π−n/2∫ℝne−iηyψηdη.(19)U3:L2ℝ2n⟶L2ℝ2n,U3fx,y=f12x+y,12x−y,for z=z1,⋯,zn∈ℂn, x=x1,⋯,xn, and y=y1,⋯,yn∈ℝn. See [1], the operator U=U3U2U1 from the space L2ℂn,dμn onto L2ℝ2n,dxdy=L2ℝn,dx⊗L2ℝn,dy satisfies that the image of the true poly-Fock space Fα2ℂn is L2ℝn,dx⊗H~α−1 and the orthogonal projection Bα of L2ℂn,dμn onto Fα2ℂn is unitary equivalent to the operator
(20)UBαU−1=I⊗P~α−1.Analogously, forα∈ℤ+n and k∈ℕ, we define the spaces
(21)H~α−1⊕=⊕λi≤αiλ∈ℤ+nH~λ−1,H~k⊕=⊕α=kα∈ℤ+nH~α−1andH~k⊕=⊕∣α∣=nα∈ℤ+nkH~α−1,with respective orthogonal projections
(22)P~α−1ψy=∑λi≤αiλ∈ℤ+nh~λ−1y∫ℝnψηh~λ−1ηdη,P~kψy=∑α=kα∈ℤ+nh~α−1y∫ℝnψηh~α−1ηdη,P~kψy=∑α=nα∈ℤ+nkh~α−1y∫ℝnψηh~α−1ηdη.If we denote byBα, Bk, and Bk the orthogonal projections of L2ℂn,dμn onto Fα2ℂn, Fk2ℂn and Fk2ℂn, respectively; thus,
(23)UBαU−1=I⊗P~α−1,UBkU−1=I⊗P~k,UBkU−1=I⊗P~k.Consider the isometric inmersionR~0,α:L2ℝn,dx⟶L2ℝ2n,dxdy, defined by R~0,αfx,y=fxh~α−1y. Whose adjoint operator R~0,α∗:L2ℝ2n,dxdy⟶L2ℝn,dx is given by R~0,α∗φx=∫ℝnφx,ηh~α−1ηdη. These operators satisfy the following relations:
(24)R~0,α∗R~0,α=I:L2ℝn,dx⟶L2ℝn,dx,R~0,αR~0,α∗=I⊗P~α−1:L2ℝ2n,dxdy⟶L2ℝ2n,dxdy.Now, we define the operatorR~α=R~0,α∗U from L2ℂn,dμn onto L2ℝn,dx, and the adjoint operator R~α∗=U∗R~0,α, which satisfy
(25)R~α∗R~α=Bα:L2ℂn,dμn⟶Fα2ℂn,(26)R~αR~α∗=I:L2ℝn,dx⟶L2ℝn,dx.For the multi-indexα=α1,⋯,αn∈ℤ+n, remember that Pα=α1⋯αn, so we introduce the isometric inmersion R~0,α:L2ℝn,dxPα⟶L2ℝn,dx⊗L2ℝn,dy, defined by
(27)R~0,αfx,y=∑λi≤αiλ∈ℤ+nfλxh~λ−1y=fxNαyT,where the Pα-tuples f=fλλi≤αi, and Nαy=h~λ−1yλi≤αi are taken over all multi-indices λ=λ1,⋯,λn whose entries are less than or equal to each of the corresponding entries of α. The adjoint operator R~0,α∗:L2ℝ2n,dxdy⟶L2ℝn,dxPα is given by
(28)R~0,α∗φx=∫ℝnφx,yh~λ−1ydyλi≤αiλ∈ℤ+n=∫ℝnφx,yNαydy.We have(29)R~0,α∗R~0,α=I:L2ℝn,dxPα⟶L2ℝn,dxPα,R~0,αR~0,α∗=I⊗P~α−1:L2ℝ2n,dxdy⟶L2ℝn,dx⊗H~α−1⊕.The operatorR~α=R~0,α∗U from L2ℂn,dμn onto L2ℝn,dxPα and its adjoint R~α∗=U∗R~0,α satisfy
(30)R~α∗R~α=Bα:L2ℂn,dμn⟶Fα2ℂn,(31)R~αR~α∗=I:L2ℝn,dxPα⟶L2ℝn,dxPα.Similarly to (27), we define the operators
(32)R~0,k:L2ℝn,dxsk⟶L2ℝ2n,dxdy,R~0,kfx,y=fxNkyT,R~0,k:L2ℝn,dxsk⟶L2ℝ2n,dxdy,R~0,kfx,y=fxNkyT.The arrangementNky is formed by the functions h~α−1y over all multi-indices α such that α=k. Analogously, the functions in Nky are indexed with the multi-indices whose absolute value are less or equal to k. Remember that sk=k−1n−1 and sk=kk−n. The adjoint operators R~0,k∗ and R~0,k∗ are defined similarly to (28). Finally, the operators R~k=R~0,k∗U, R~k=R~0,k∗U, and its adjoint operators satisfy
(33)R~k∗R~k=Bk:L2ℂn,dμn⟶Fk2ℂn,R~kR~k∗=I:L2ℝn,dxsk⟶L2ℝn,dxsk,R~k∗R~k=Bk:L2ℂn,dμn⟶Fk2ℂn,R~kR~k∗=I:L2ℝn,dxsk⟶L2ℝn,dxsk.


## 3. Toeplitz Operators with Horizontal Symbols

In this section, we define Toeplitz operators with certain class of symbols acting on the poly-Fock, true poly-Fock, and homogeneously poly-Fock spaces overℂn. And, we prove that this operators are unitary equivalent to certain multiplication operators. Let az=ax1,⋯,xn be a function in L∞ℝn,dx depending only on x=Rez1,⋯,Rezn, we call to this kind of functions horizontal symbols. Henceforth, α∈ℤ+n denote a fixed multi-index and k∈ℕ a fixed natural number.Definition 5.
Letax be a horizontal symbol. The Toeplitz operator with symbol ax, acting on the true poly-Fock space (or poly-Fock space) of order α is defined as
(34)Tα,a:φ∈Fα2ℂn↦Bαaφ∈Fα2ℂn,(35)Tα,a:φ∈Fα2ℂn↦Bαaφ∈Fα2ℂn.
Similarly, the Toeplitz operator with symbolax, acting on the homogeneously poly-Fock space (or poly-Fock space) of order k is defined as
(36)Tk,a:φ∈Fk2ℂn↦Bkaφ∈Fk2ℂn,(37)Tk,a:φ∈Fk2ℂn↦Bkaφ∈Fk2ℂn.The following theorem characterizes the Toeplitz operators with horizontal symbols acting on the true poly-Fock spaceFα2ℂn.Theorem 6.
Letax∈L∞ℝn,dx be a horizontal symbol, then the Toeplitz operator Tα,a acting on Fα2ℂn is unitary equivalent to the multiplication operator γα,aI=R~αTα,aR~α∗ acting on L2ℝn,dx where the function γα,a is given by
(38)γα,ax=∫ℝnax+y2h~α−1yh~α−1y2dy,and h~α−1y is defined in (8).Proof.
Remember thatR~α=R~0,α∗U and using (25) and (26), we obtain
(39)R~αTα,aR~α∗=R~αBαaBαR~α∗=R~αR~α∗R~αaR~α∗R~αR~α∗=R~αaR~α∗=R~0,α∗U3U2U1axU1−1U2−1U3−1R~0,α=R~0,α∗U3axU3−1R~0,α=R~0,α∗ax+y2R~0,α.
Explicitly for a functionf∈L2ℝn, we have
(40)R~0,α∗ax+y2R~0,αfx=∫ℝnax+y2fxh~α−1y2dy=γα,ax·fx.
We call to the functionγα,ax the αth spectral function for the Toeplitz operator with horizontal symbol a acting on the true poly-Fock space Fα2ℂn.
Naturally, we can extend the above result to the case of the Toeplitz operator with horizontal symbols acting on the poly-Fock spaceFα2ℂn.Theorem 7.
Letax∈L∞ℝn,dx be a horizontal symbol; thus, the Toeplitz operator Tα,a acting on Fα2ℂn is unitary equivalent to the multiplication operator γα,axI=R~αTα,aR~α∗, acting on L2ℝn,dxPα, where the matrix γα,a is given by
(41)γα,ax=∫ℝnax+y2NαyTNαydy.
That is, each component function is equal to(42)γα,aλμx=∫ℝnax+y2h~λ−1yh~μ−1ydy,with λ,μ∈ℤ+n such that λi,μi≤αi and Nαy is defined in (27).Proof.
SinceR~α=R~0,α∗U and using (30) and (31), we obtain
(43)R~αTα,aR~α∗=R~αBαaBαR~α∗=R~αR~α∗R~αaR~α∗R~αR~α∗=R~αaR~α∗=R~0,α∗U3U2U1axU1−1U2−1U3−1R~0,α=R~0,α∗U3axU3−1R~0,α=R~0,α∗ax+y2R~0,α.
Calculating for a functionf∈L2ℝn,dxPα(44)R~0,α∗ax+y2R~0,αfx=R~0,α∗ax+y2fxNαyT=∫ℝnax+y2fxNαyTNαydy=∫ℝnax+y2NαyTNαyfxdy=γα,axfx.
We have the next two theorems, whose proofs are analogous to the above one.Theorem 8.
For a horizontal symbolax∈L∞ℝn,dx, the Toeplitz operator Tk,a acting on the homogeneously poly-Fock space Fk2ℂn is unitary equivalent to the multiplication operator γk,axI=R~kTk,aR~k∗, acting on L2ℝn,dxsk, where
(45)γk,ax=∫ℝnax+y2NkyTNkydy.Theorem 9.
For a horizontal symbolax∈L∞ℝn,dx, the Toeplitz operator Tk,a acting on the poly-Fock space Fk2ℂn is unitary equivalent to the multiplication operator γk,axI=R~kTk,aR~k∗, acting on L2ℝn,dxsk, where
(46)γk,ax=∫ℝnax+y2NkyTNkydy.We call to the matricesγα,ax, γk,ax, and γk,ax the spectral matrices correspondent to the Toeplitz operator with horizontal symbol ax, acting on the poly-Fock space Fα2ℂn, on the homogeneously poly-Fock space Fk2ℂn, and on the poly-Fock space Fk2ℂn, respectively.Remark 10.
The components of the spectral matricesγα,ax,γk,ax, and γk,ax can be expressed as a convolution of functions a¯∗h~λh~μ, where a¯x=ax/2∈L∞ℝn. Since h~λh~μ∈L1ℝn, from [7], p. 283, we guaranteed that they belong in the set of uniformily continuous functions Cbℝn.


## 4. TheC∗-Algebras Generated by Toeplitz Operators with Extended Horizontal Symbols

In this section, we introduce the concept of extended horizontal symbol and we describe theC∗-algebras generated by Toeplitz operators with these symbols acting on the poly-Fock spaces and on the homogeneously poly-Fock spaces. Following the terminology and the notation introduced in [8], Section 3, we have the following.Definition 11.
Letax∈L∞ℝn be a horizontal symbol. We say that ax is an extended horizontal symbol if there exists a function a∞x∈CSn−1 such that
(47)limR⟶∞supx>Rax−a∞xx=0.
We denote byHSℝn the set of extended horizontal symbols. We note that HSℝn equipped with the supremum norm is a C∗-subalgebra of L∞ℝn.The compact of maximal ideals of theC∗-algebra HSℝn coincides with the compactification of ℝn, denoted by ℝ~n=ℝn∪S∞n−1, obtained by adding an “infinitely far” n-sphere S∞n−1. This compact space is isomorphic to Dn¯. We can identify the elements s∞∈S∞n−1 with the points s∈Sn−1 as follows. For every extended horizontal symbol a∈HSℝn, we have
(48)limt⟶∞ats=as∞=a∞s.We identify the extended horizontal symbolsax with its extensions to the complex space ℂn, where x=Rez.The following lemma shows that the different spectral matricesγα,ax, γk,ax, and γk,ax, corresponding to Toeplitz operators with extended horizontal symbol ax, posses a limit value to infinity in any direction. We write γ□,ax to refer to any of this spectral matrices.Lemma 12.
Letax be an extended horizontal symbol and let x0∈Sn−1. Then the spectral matrix γ□,ax satisfies
(49)limt⟶∞γ□,axtx0=a∞x0I.Proof.
We apply the dominated convergence theorem. Letλ,μ∈ℤ+n be two multi-indices corresponding to some entry of the spectral matrix. For each m∈ℕ, we consider the function Fm:ℝn⟶ℝ defined by
(50)Fmy=amx0+y2h~λ−1yh~μ−1y.
Sinceh~λ−1y,h~μ−1y∈L2ℝn, and ax∈L∞ℝn we have Fmy∈L1ℝn. Note that the integrable function a∞h~λ−1yh~μ−1y limits to Fmy for all m∈ℕ. Since the function a∞ is continuous, we have
(51)limm⟶∞a∞x0+1/myx0+1/my=a∞x0,and using (48)
(52)limm⟶∞limt⟶∞atx0+1/myx0+1/my=limm⟶∞limt⟶∞atx0+1/my2=a∞x0.
We can take this limit along the linet=m; thus,
(53)limm⟶∞amx0+y2=a∞x0,and Fmy⟶a∞xh~λ−1yh~μ−1y when m⟶∞. Therefore,
(54)limm⟶∞∫ℝnamx0+y2h~λ−1yh~μ−1ydy=a∞x∫ℝnh~λ−1yh~μ−1ydy=a∞xδλμ.
Letα=α1,⋯,αn∈ℤ+n be a fixed multi-index and k∈ℕ be a fixed natural number.Definition 13.
We introduce the followingC∗ algebras, which are very useful to our study
(i)
DenoteGαH=γα,a:ax∈HSℝn to the set of all horizontal spectral functions(ii)
DenoteGαH=γα,a:ax∈HSℝn to the set of all horizontal spectral matrices of order α(iii)
DenoteGkH=γk,a:ax∈HSℝn to the set of all horizontal spectral matrices of order exactly k(iv)
DenoteGkH=γk,a:ax∈HSℝn to the set of all horizontal spectral matrices of order at most k(v)
Denote byT∞α the C∗-algebra generated by the set of Toeplitz operators Tα,a acting on the true poly-Fock space Fα2ℂn, with ax∈HSℝn(vi)
Denote byT∞α the C∗-algebra generated by the set of Toeplitz operators Tα,a acting in the poly-Fock space Fα2ℂn, with ax∈HSℝn.(vii)
Denote byT∞k the C∗-algebra generated by the set of Toeplitz operators Tk,a acting on the homogeneously poly-Fock space Fk2ℂn, with ax∈HSℝn(viii)
Denote byT∞k the C∗-algebra generated by the set of Toeplitz operators Tk,a acting in the poly-Fock space Fk2ℂn, with ax∈HSℝnWe have the following results.Corollary 14.
TheC∗-algebra T∞α is isometrically isomorphic to the C∗-algebra GαH generated by GαH.Corollary 15.
TheC∗-algebra T∞α is isometrically isomorphic to the C∗-algebra GαH generated by GαH.Corollary 16.
TheC∗-algebra T∞k is isometrically isomorphic to the C∗-algebra GkH generated by GkH.Corollary 17.
TheC∗-algebra T∞k is isometrically isomorphic to the C∗-algebra GkH generated by GkH.Now, we describe theC∗-algebras GαH, GkH, and GkH, generated by the different spectral matrices. First, we start with GαH. Consider the C∗-algebra defined by Cα=MPαℂ⊗Cℝn∪S∞n−1, which consists of the algebra of all Pα×Pα matrices with entries in Cℝn∪S∞n−1, where Pα=α1⋯αn. Now, we introduce the C∗-subalgebra Dα given by
(55)Dα=M∈Cα:limt⟶∞Mtx=Mx∞=ℂI,forallx∈Sn−1.We note thatGαH is a C∗-subalgebra of Dα. In fact, we prove that GαH=Dα. For this, we use a Stone-Weirstrass theorem. We need to show that GαH separates the pure states of Dα.SinceDα is a C∗-bundle, the set of all its pure states is completely determined by the pure states on the fibers:
(56)Dαx=MPαℂ,ifx∈ℝn,ℂ,ifx=x∞∈S∞n−1.So each pure state ofDα has the form
(57)fM=fxMx,M∈Dα,where x∈ℝn∪S∞n−1 and fx is a pure state of Dαx. Every pure state in the matrix algebra MPαℂ is given by a functional fv defined as
(58)fvQ=Qv,v,Q∈MPαℂ,with v∈SPα=z∈ℂPα:z=1. Moreover, if v,w∈SPα such that fv=fw; thus, v=tw where t∈ℂ and t=1, see [9] for more details.In consequence, the set of all pure states ofDα consists of all functional of the form
(59)fx,vM=Mxv,v,M∈Dα,with x∈ℝn∪S∞n−1 and v∈SPα.In the cases of theC∗-algebras GkH and GkH, we consider the C∗-algebras Ck=Mskℂ⊗Cℝn∪S∞n−1 and Ck=Mskℂ⊗Cℝn∪S∞n−1. And their corresponding C∗-subalgebras Dk and Dk. For this two C∗-subalgebras, the pure states are determined in a similar way to (59). Remember that sk=k−1n−1 and sk=kk−n.Now to fixing ideas, we return to the previous caseGαH; the other ones are totally analogous, and we analyzed them at the end of this section.For each elementx∞∈S∞n−1, we have only one pure state for any v,w∈SPα, that is, fx∞,v=fx∞,w. To separate the pure states corresponding to two different elements x0,∞ and x1,∞ in S∞n−1, using the identification given by (48), we note that the corresponding elements x0,x1∈Sn−1 differ at least one coordinate. Suppose that the jth coordinate of this vectors is different, that is, x0,j≠x1,j. Thus, we consider the horizontal extended symbol Cj:ℝn⟶ℂ defined by
(60)Cjx=0,ifx=0∈ℝn,xjx,inotherwise.Forx=x1,⋯,xn∈ℝn with x=1, we have C∞jx=limt⟶∞Cjtx=xj. Clearly, C∞j is a continuous function. Now, consider the spectral matrix γα,Cj, for every x∞, we have
(61)fx∞,vγα,Cj=γα,Cjx∞v,v=C∞jxIv,v=xj.Hence, forx0,∞≠x1,∞, we have
(62)fx0,∞,vγα,Cjx≠fx1,∞,vγα,Cjx.Thus, the spectral matrixγα,Cjx separates the corresponding pure states.In the case when we have the pure states corresponding to the pointsx0∈ℝn and x1,∞∈S∞n−1, we consider the set 0,p=0,p1×⋯×0,pn with p=p1,⋯,pn∈ℝ+n, and the function cx=χ0,px. We have c∞x≡0. We write γα,px instead of γα,cx. Notice that
(63)γα,px=∫−xn2pn−xn⋯∫−x12p1−x1NαyNαyTdy1⋯dyn=∫−x,2p−xNαyNαyTdy.Forx0∈ℝn, we have
(64)fx0,vγα,p=γα,px0v,v=∫−x,2p−xNαyNαyTdyv,v=∫−x,2p−xNαyNαyTv,vdy=∫−x,2p−xv,NαyNαy,vdy=∫−x,2p−xv,Nαy2dy.Note thatfx0,vγα,p>0 because <v,Nαy>2≥0, except in a set of measure zero. On the other hand, if x1∈Sn−1 is the corresponding element of x1,∞∈S∞n−1, we have
(65)fx1,∞,vγα,p=γα,px1,∞v,v=c∞x1Iv,v=0.Therefore, the spectral matrixγα,p separates the pure states of the points x0 and x1,∞.To separate the pure states corresponding to two pointsx0,x1∈ℝn, we consider again the extended horizontal symbol cx=χ0,px. From (64), we define the function hvy=<v,Nαy>2. We can express this function as hvy=qvye−y12+⋯+yn2, with
(66)qvy=∑λ∈ℤ+nλi≤αivλ2λλ!πn2−1/2H~λ−1y2,whereH~λ−1y is the product of the n one-dimensional Hermite polynomials Hλ1−1y1,⋯,Hλn−1yn, so qvy is a nonnegative-valued polynomial of degree at most 2∣α∣−2n.The following lemma provides us a tool to prove that theC∗-algebra GαH separates the pure states of Dα of the form fx0,v,fx1,w, where x0≠x1 and v,w∈SPα.Lemma 18.
We assume thatv,w∈SPα,x0,x1∈ℝn and γα,p∈GαH with p∈ℝ+n. Iffx0,vγα,px=fx0,wγα,px for all vector p, then x0=x1. Moreover, v,Nαy2=w,Nαy2for ally∈ℝn.Proof.
The hypothesisfx0,vγα,px=fx1,wγα,px is equivalent to the following equation:
(67)∫−x0,2p−x0qvye−y12+⋯+yn2dy=∫−x1,2p−x1qwye−y12+⋯+yn2dy.
Taking the partial derivative∂n/∂p=∂n/∂pn⋯∂p1, we obtain
(68)qv2p−x0e−2p1−x0,12+⋯+2pn−x0,n2=qw2p−x1e−2p1−x1,12+⋯+2pn−x1,n2,thus,
(69)qv2p−x0=qw2p−x1e∑j=1n2pj−x0,j2−2pj−x1,j2=qw2p−x1e∑j=1n22pjx1,j−x0,j+x0,j2−x1,j2.
Sinceqv and qw are polynomials, this fact implies that the exponential part in the above equation is constant for all p∈ℝ+n. Hence, x0,j=x1,j for all j; therefore, x0=x1. Using this fact, it is clear that qv2p−x0=qw2p−x1, for all p, that is, v,Nαy2=w,Nαy2 for all y∈ℝn.
As consequence of the above lemma, ifx0≠x1 and v,w∈SPα, then there exists p0∈ℝ+n such that fx0,vγα,p0x≠fx1,wγα,p0x; hence, the spectral matrix γα,p0 separates the pure states fx0,v and fx1,w.
To complete the proof of the fact that theC∗-algebra GkH separates all the pure states of D, only missing step is to separate the pure states of the forms fx,v and fx,w, where v,w∈SPα and x∈ℝn. For this, we need to deduce some useful facts before.For the multi-indexα=α1,⋯,αn, we consider Nαy=h~λ−1yλi≤αi. From (8), we have h~λ−1y=hλ1−1y1⋯hλn−1yn, where hλi−1yi is a one-dimensional Hermite function. We can consider that all correspondent one-dimensional Hermite’s polynomials Hλi−1yi, whose degrees are equal to λi−1, are monic polynomials. Hence, for every multi-index λ such that λi≤αi, we can write
(70)h~λ−1y=2λ−1/2λ−1!πn−1/2e−y2/2Hλ1−1y1⋯Hλn−1yn.Now, fory∈ℝ−0, we construct the vector y⟵∈ℝn with the form
(71)y⟵=yα2⋯αn,yα3⋯αn,⋯,yαn−1·αn,yαn,y.Evaluating this vector in Hermite’s polynomials corresponding to the multi-indexλ=λ1,⋯,λn(72)Hλ1−1yα2⋯αn·Hλ2−1yα3⋯αn⋯Hλn−1−1yαn·Hλn−1y,we obtain a polynomial dependents on only one variable, whose degree we can calculate with the equation:
(73)pλ=∑j=1n−1λj−1αj+1⋯αn+λn−1.Consequently, evaluatingy⟵ in h~λ−1, we can write
(74)h~λ−1y⟵=2λ−1/2λ−1!πn−1/2e−∥y⟵∥22ypλ+aλ,1ypλ−1+⋯+aλ,pλy+1.From (73), we notice that for two different multi-indices λ and μ, the corresponding degrees satisfy pλ≠pμ. Moreover, for 1n=1,⋯,1, we have p1n=0. And for α=α1,⋯,αn,
(75)pα=∑j=1n−1αj−1αj+1⋯αn+αn−1=α1⋯αn−1=Pα−1.Therefore, the multi-indicesλ∈ℤ+n such that λi≤αi for every i generate different polynomials of degrees between 0 and Pα−1. Now, for each of these multi-indices λ, we consider vectors yλ⟵ defined by (71), and we define the matrix N whose dimension is Pα×Pα and the λth row is equal to Nαyλ⟵. Since the components of Nαy are sorted ascending by the lexicographic order, we claim that the matrix N has the form:
(76)N=Cα·D·1y1n+a0y1n2+b1y1n+b0⋯y1nPα−1+⋯⋮⋮⋮⋮1yα+a0yα2+b1yα+b0⋯yαPα−1+⋯,where
(77)Cα=∏λ∈ℤ+nλi≤αi2λ−1/2λ−1!πn−1/2,D=diage−y⟵μ2/2μ∈ℤ+nμi≤αi.We can calculate the determinant ofN using multilineality and the VandermondeÂ’s formula; we obtain
(78)detN=∏λ∈ℤ+nλi≤αi2∣λ−1∣/2λ−1!πn−1/2e−∑μ∥yμ∥2/2∏γ,δ∈ℤ+n1n≤γ<δ≤αyδ−yγ≠0.Example 1.
Considern=3 and the multi-index α=2,4,1. We have Pα=8 and the multi-indices, arranged with the lexicographic order, whose coordinates are less or equal to the corresponding coordinates of α are
(79)1n=λ1=1,1,1,λ2=1,2,1,λ3=1,3,1,λ4=1,4,1,λ5=2,1,1,λ6=2,2,1,λ7=2,3,1,λ8=2,4,1=α.
In this case, the vectory⟵=y4,y,y. From (73), we can calculate the different degrees, for example for λ7=2,3,1,
(80)pλ7=1·4+2·1+1−1=6.
And for the rest of multi-indices, we have(81)pλ1=0,pλ2=1,pλ3=2,pλ4=3,pλ5=4,pλ6=5,pλ7=6,pλ8=7.
Now, we can continue with the separation of the pure states.Lemma 19.
Givenv,w∈SPα and x∈ℝn being fixed, consider the spectral matrices γα,p,γα,r. If fx,vγα,p=fx,wγα,r, for all p,r∈ℝ+n then v=tw, with t∈ℂ and t=1.Proof.
From Lemma18, we have <v,Nαy>2=<w,Nαy>2 for all y∈ℝn; thus, there exists a function θ:ℝn⟶ℂ such that
(82)v,Nαy=eiθyw,Nαy.
Foru∈SPα, we define the function Hu:ℝ+2n⟶ℂ given by
(83)Hup,r=fx,uγα,pγα,r=∫−x2p−xu¯TNαyNαyTdy∫−x2r−xNαyNαyTudy.
Without loss of generality, we can assume thatx=0,⋯,0∈ℝn, just to simplify the calculations. Taking the derivative of order 2n of the function Hu, with respect to p and r, we obtain
(84)∂2nHu∂p∂rp2,r2=u¯TNαpNαpTNαrNαrTu=Nαp,uNαr,Nαpu,Nαr.
The hypothesisHv=Hw implies that ∂2nHv/∂p∂r=∂2nHw/∂p∂r. Using (82) and (84),
(85)Nαp,wNαr,Nαpw,Nαr=eiθr−θpNαp,w××Nαr,Nαpw,Nαr,for all p,r∈ℝ+n. Clearly, there exists r0∈ℝ+n such that w,Nαr0≠0. Thus,
(86)1−eiθr0−θpNαp,wNαr0,Nαp=0.
Notice thatNαp,wNαr0,Nαp is a nonzero polynomial with respect to p; thus, the above equation implies that the function θ is constant. From (82), we obtain that v−eiθ0w,Nαy=0, for all y∈ℝn. Using (78), we have v−eiθ0w=0 and v=eiθ0w.Finally, we consider the case of theC∗-algebras GkH and GkH. Since the proof of the Lemma 19 is independent of the dimension Pα and the nature of the multi-indices λ and μ, we can obtain the following analogous results.Lemma 20.
Forv,w∈Ssk,x0,x1∈ℝn and γk,p∈GkH with p∈ℝ+n. If fx0,vγk,px=fx0,wγk,px for all vector p, then x0=x1. Also, v,Nky2=w,Nky2 for all y∈ℝn.Lemma 21.
Forv,w∈Ssk,x0,x1∈ℝn and γk,p∈GkH with p∈ℝ+n. If fx0,vγk,px=fx0,wγk,px for all vector p, then x0=x1. Also, v,Nky2=w,Nky2 for all y∈ℝn.On the other hand, we consider theC∗-algebra GkH and the multi-index
(87)kn=k−n−1,⋯,k−n−1.Ifα=α1,⋯,αn is a multi-index such that ∣α∣=k then all its components satisfy αi≤k−n−1. So, if we construct the Pkn×Pkn invertible matrix N as in (76), corresponding to the multi-index kn, and apply it to the vector
(88)0,⋯,v−eiθ0w,⋯,0,where the coordinates of v−eiθ0w∈Ssk occupy the same positions that the multi-indices whose absolute value is equal to k in the matrix N. Thus, we can prove the following.Lemma 22.
Givenv,w∈Ssk and x∈ℝn being fixed, consider the spectral matrices γk,p,γk,r. If fx,vγk,p=fx,wγk,r, for all p,r∈ℝ+n then v=tw, with t∈ℂ and t=1.Analogously, for the case of theC∗-algebra GkH and the set of multi-indices with absolute value is less than or equal to k, we have the following.Lemma 23.
Givenv,w∈Ssk and x∈ℝn being fixed, consider the spectral matrices γk,p,γk,r. If fx,vγk,p=fx,wγk,r, for all p,r∈ℝ+n then v=tw, with t∈ℂ and t=1.For the noncommutative Stone-Weierstrass conjecture, letB be a C∗-subalgebra of a C∗-algebra A, and suppose that B separates all the pure states of A (and 0 if A is nonunital). Then, A=B.In [10], Kaplansky proved this conjecture for a C∗-algebra type I. In consequence, we prove that the algebra GαH is equal to Dα. From Corollary 15, we have that the algebra of Toeplitz operators T∞α is isometric and isomorphic to the algebra Dα. Analogously, applying the Corollary 16 and the Corollary 17, we have that T∞k and T∞k are isometric and isomorphic to Dk and Dk, respectively. In summary, we have the following results.Theorem 24.
TheC∗-algebra T∞α is isomorphic and isometric to the C∗-algebra Dα. The isomorphism is given by
(89)T∞α:Tα,a↦SymTα,ax=γα,ax,where γα,ax is given in (41).Theorem 25.
TheC∗-algebra T∞k is isomorphic and isometric to the C∗-algebra Dk. The isomorphism is given by
(90)T∞k:Tk,a↦SymTk,ax=γk,ax,where γk,ax is given in (45).Theorem 26.
TheC∗-algebra T∞k is isomorphic and isometric to the C∗-algebra Dk. The isomorphism is given by
(91)T∞k:Tk,a↦SymTk,ax=γk,ax,where γk,ax is given in (46).Corollary 27.
TheC∗-algebra T∞α is isomorphic and isometric to the commutative C∗-algebra Cℝn∪S∞n−1. The isomorphism is given by
(92)T∞α:Tα,a↦SymTα,ax=γα,ax,where γα,ax is given in (38).


## 5. Toeplitz Operators withL-Invariant Symbols

In this section, we introduce the extended Lagrangian symbols, and we prove that theC∗-algebra generated by Toeplitz operators with this kind of symbols acting on the homogeneously poly-Fock space is isomorphic and isometric to the C∗-algebra generated by Toeplitz operators with extended horizontal symbols acting on this same space.We consider the standard symplectic formω0 of ℂn=ℝ2n given by ω0z,w=Jz·w,forallz,w, where
(93)J=0In−In0.Recall that an-dimensional subspace L⊂ℝ2n is called a Lagrangian plane if for every z,w∈L it satisfy ω0z,w=0. Clearly, iℝn=0×ℝn is a Lagrangian plane. We denote by Lag2n,ℝ the set of all Lagrangian planes in ℝ2n. If we consider the transitive group action of U2n,ℝ onto Lag2n,ℝ defined by
(94)X,L↦XL,we have that for every Lagrangian plane L there is an unitary matrix X such that XL=iℝn. For more details, see [11], Proposition 43. Since the unitary group U2n,ℝ is isomorphic to Un,ℂ, each Lagrangian plane L can be identified with a subspace of ℂn; abusing the notation, we denote this subspace with L too.LetL be a Lagrangian plane, we say that a function φ∈L∞ℂn is L-invariant or Lagrangian invariant if for every h∈L it satisfies
(95)φz−h=φz,foralmostallz∈ℂn,so we can consider it like a function depending only on the elements of L∁.In [5], Esmeral and Vasilevski introduced the concept of L-invariant functions and they provided the following criterion for a function to be so.Lemma 28.
Consider a Lagrangian planeL and X∈Un,ℂ such that XL=iℝn. Then, a function φ∈L∞ℂn is L-invariant if and only if there exists a∈L∞ℝn such that
(96)φX∗z=aRez1,⋯,Rezn,foralmostallz∈ℂn.
Moreover, they established the following result.Proposition 29.
TheC∗-algebra generated by Toeplitz operators with horizontal symbols acting on the Fock space F2ℂn is unitary equivalent to the C∗-algebra generated by Toeplitz operators with L-invariant symbols.
For this, they introduced the operatorVX:L2ℂn,dμn⟶L2ℂn,dμn defined by
(97)VXfz=fX∗z,withX∈U2n,ℝsuchthatXL=iℝn.
SinceX∗=X−1, this operator is unitary and VX∗=VX−1. It too satisfies
(98)VXKz=KXz,where Kz is the reproducing kernel of F2ℂn in the point z.
In the case of the poly-Fock space and the true poly-Fock space, the above result could fail, because for some multi-indexα and some unitary matrix X; the spaces Fα2ℂn and Fα2ℂn might not be invariant under the operator VX.Example 1.
Considern=2, α=2,1. Using (15), we have
(99)ψz=z1¯z1+z2∈Fα2ℂn,however, if X∗=aij with i,j=1,2 is an unitary matrix, it is clear that
(100)ψX∗z=w1¯w1+w2∉Fα2ℂn,where w1=a11z1+a12z2, w2=a21z1+a22z2.This is the main motivation for which we consider the homogeneously poly-Fock spaceFk2ℂn and the poly-Fock space Fk2ℂn for k∈ℕ.Note that forL∈ Lag2n,ℝ, X∈U2n,ℝ such that XL=iℝn, and VX defined by (97), using the explicit form of the elements in the true poly-Fock space given by (15), from [6], Proposition 2.7, we have that the homogeneously poly-Fock space Fk2ℂn and the poly-Fock space Fk2ℂn are invariant under VX.Now, we define the extended Lagrangian symbols; these kind of symbols is related with the extended horizontal symbols as follows.Definition 30.
ForL∈Lag2n,ℝ and X∈U2n,ℝ such that XL=iℝn, equivalently XL∁=ℝn and X∗ℝn=L∁, consider the following diagram.
We say that theL-invariant symbol φ is an extended Lagrangian symbol or an extended L-invariant symbol if its pullback by X∗ is an extended horizontal symbol. In other words, if the function aRez=φX∗z given in (96), it is an extended horizontal symbol.According to the above diagram, forz∈L∁, there exists w∈ℂn such that z=X∗Rew and z=Rew. If φ is an extended Lagrangian symbol, using (96) and (48), we have
(101)limt⟶∞φtz=limt⟶∞φtX∗Rew=limt⟶∞atRew=a∞RewRew=a∞Xzz.If we define the continuous functionφ∞:S2n−1∩L∁⟶ℂ given by
(102)φ∞z=a∞Xz,we have for z∈L∁(103)limt⟶∞φtz=φ∞zz.This functionφ∞ is invariant under translations by Lagrangians elements whose norm is equal to 1. Let h∈L such that h=1, so Xh∈iℝn and Xh=1; thus, for z∈S2n−1∩L∁, we have
(104)φ∞z=a∞Xz=a∞Xz+Xh=a∞Xz+h=φ∞z+h.Lemma 31.
Consider the unitary operatorVXforX∈U2n,ℝ. IfKkz,wdenotes the reproducing kernel of the true homogeneously poly-Fock spaceFk2ℂnthenVXKvk=KXvk.Proof.
Letφz∈Fk2ℂn. Using the reproducing property, we can express
(105)φz=1πn∫ℂnφwKkz,w¯e−w2dw.
ApplyVX(106)VXφz=1πn∫ℂnφX∗wKkz,w¯e−w2dw,since X∗ is unitary and taking v=X∗w, we have
(107)VXφz=1πn∫ℂnφvKkz,Xv¯e−v2dv.
On the other hand,(108)φX∗z=1πn∫ℂnφvKkX∗z,v¯e−v2dv,for the uniqueness of the reproducing kernel, we have Kkv,X∗z=KkXv,z; therefore VXKvk=KXvk.Corollary 32.
The reproducing kernelKkz,w of the poly-Fock space Fk2ℂn satisfies VXKvk=KXvk.
Now, considera∈L∞ℂn and X∈U2n,ℝ. Using the Lemma 31, we obtain that the Toeplitz operator Ta acting on the true homogeneously poly-Fock space Fk2ℂn is unitary equivalent to
(109)VX−1TaVXφz=VX−1TaVXφ,Kzk=TaVXφ,VXKzk=aVXφ,KXzk=aXφ,Kzk=TaXφz,where aXz=aXz. Analogously, by Corollary 32, the Toeplitz operator Ta acting in the poly-Fock space Fk2ℂn is unitary equivalent to TaX.
Using the above results, we obtain the following generalizations of Proposition29.Theorem 33.
TheC∗-algebra T∞k generated by Toeplitz operators with extended horizontal symbols acting on the true homogeneously poly-Fock space Fk2ℂn is unitary equivalent to the C∗-algebra TL,∞k generated by Toeplitz operators with extended L-invariant symbols.Theorem 34.
TheC∗-algebra T∞k generated by Toeplitz operators with extended horizontal symbols acting in the poly-Fock space Fk2ℂn is unitary equivalent to the C∗-algebra TL,∞k generated by Toeplitz operators with extended L-invariant symbols.Finally, using Theorems (25) and (26), we have the following.Corollary 35.
TheC∗-algebra TL,∞k generated by Toeplitz operators with extended L-invariant symbols acting on the homogeneously poly-Fock space Fk2ℂn is isomorphic and isometric to the C∗-algebra Dk.Corollary 36.
TheC∗-algebra TL,∞k generated by Toeplitz operators with extended L-invariant symbols acting on the poly-Fock space Fk2ℂn is isomorphic and isometric to the C∗-algebra Dk.



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