# Toeplitz Operators with Horizontal Symbols Acting on the Poly-Fock Spaces

Authors: Armando Sánchez-Nungaray; Carlos González-Flores; Raquiel Rufino López-Martínez; Jorge Luis Arroyo-Neri
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We describe theC⁎-algebra generated by the Toeplitz operators acting on each poly-Fock space of the complex plane C with the Gaussian measure, where the symbols are bounded functions depending only on x=Rez and have limit values at y=-∞ and y=∞. The C⁎ algebra generated with this kind of symbols is isomorphic to the C⁎-algebra functions on extended reals with values on the matrices of dimension n×n, and the limits at y=-∞ and y=∞ are scalar multiples of the identity matrix.


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## Body

## 1. Introduction

Recall that then poly-Fock space is denoted by Fn2(C)⊂L2(C,e-|z|2) and consists of the n-analytic functions which satisfy the equation (1)∂∂z¯nφ=12n∂∂x+i∂∂ynφ=0.The n true poly-Fock space is denote by F(n)2(C), which consists of the all true-n-analytic functions; i.e., (2)Fn2C=Fn2C⊖Fn-12C,for n≥1, and F(0)2(C)={0}.It is clear thatF12(C) is the classical Fock space on the complex plane C, which is also denoted by F2(C).In [1], N. Vasilevski proved that L2(C) has a decomposition as a direct sum of the n true poly-Fock and n true anti-poly-Fock spaces: (3)L2C,e-z2=⨁k=1∞Fn2CMoreover, they proved that the spaces F(n)2(C) are isomorphic and isometric to L2(R)⊗Hn-1, where Hn-1 is the one-dimensional space generated by Hermite function of order n-1. Finally, they found the explicit expressions for the reproduction kernels of all these function spaces.In [2], K. Esmeral and N. Vasilevski introduced the so-called horizontal Toeplitz operators acting on the Fock space and give an explicit description of the C∗-algebra generated by them. They showed that any Toeplitz operator with L∞-symbol, which is invariant under imaginary translations, is unitarily equivalent to the multiplication operator by its “spectral function”. They stated that the corresponding spectral functions form a dense subset in the C∗-algebra of bounded uniformly continuous functions with respect to the standard metric on R.The Toeplitz operators acting on spaces of polyanalytic functions have been object of study of several authors in different direction. For example, in [3], Sánchez-Nungaray and Vasilevski studied Toeplitz operators with pseudodifferential symbols acting on poly-Bergman spaces upper half plane. A different approach by Hutník, Maximenko, and Mišková in [4] considers Toeplitz Localization operators on the space of Wavelet transform or the space of short-time Fourier transform. They studied these operators with symbols that just depend on the first coordinate in the phase space, which are unitary equivalent to multiplication operators of certain specific functions “spectral functions”. In particular, the poly-Bergman spaces are spaces of Wavelet transform which is related to Laguerre functions, and the poly-Fock spaces are spaces of short-time Fourier transform which is related to Hermite functions.In [5], J. Ramírez-Ortega and A. Sánchez-Nungaray described the C∗-algebra generated by the Toeplitz operators with bounded vertical symbols and acting over each poly-Bergman space in the upper plane An2(Π). They considered bounded vertical symbols that have limit values at y=0,∞ and prove that the C∗-algebra generated by the Toeplitz operator acting on An2(Π) with this kind of symbols is isomorphic and isometric to the C∗-algebra of matrix-valued functions of the compact [0,∞]. Similar result can be found in [6], where M. Loaiza and J. Ramírez-Ortega gave an analogous description to the above for the C∗-algebra generated by the Toeplitz operators with bounded homogeneous symbols acting over each poly-Bergman space in the upper plane.The main result of this paper is the classifiedC∗-algebra generated by the Toeplitz operators with bounded vertical symbols with limits at -∞ and ∞ acting over poly-Fock space in the complex plane.This paper is organized as follows. In Section2 we introduce preliminary results about the n-polyanalytic function spaces and their relationship with the Hermite polynomials. In Section 3 we prove that every Toeplitz operator with bounded horizontal symbol a(z) acting on Fock space is unitary equivalent to a multiplication operator γn,a(x)I acting on (L2(R+))n, where γn,a(x) is a continuous matrix-valued function on (-∞,∞). Finally, in Section 4, we describe the pure states of the algebra (4)D=M∈MnC⊗CR¯:M-∞,M∞∈CI.We prove that the C∗ algebra T-∞,∞(n) generated by Toeplitz operator with bounded vertical symbols that have limit values at y=-∞,∞ acting on Fock space is isomorphic and isometric to the C∗-algebra D.


## 2. Poly-Fock Space on the Complex Plane

In this work we use the following standard notation:z=x+iy∈C, with the usual complex conjugation z¯; thus z2=z·z. The Gaussian measure on C is given by (5)dμz=1πe-z·z¯dvzwhere dv(z)=dxdy is the usual Euclidean measure on R2=C.The Hilbert spaceL2(C,dμ)=L2(C,e|z|2) of dμ(z) square integrable functions on C with the inner product (6)f,g=∫Cfzgz¯dμz.The closed subspaceF2(C) of L2(C,e|z|2) consisting of all analytic functions is called the Fock or Segal-Bargmann space. Also, the Fock space F2(C) can be defined as the closure of the set of all smooth functions satisfying the equation ∂¯ϕ=0. Similarly, given a natural number k, the k poly-Fock space Fk2(C) is the closure of the set of all smooth functions in L2(C,e|z|2) satisfying ∂¯nϕ=0.Recall that the Hermite polynomialHn(y) of degree n is defined by (7)Hny=-1ney2dndyne-y2=n!∑m=0n/2-1m2yn-2mm!n-2m!,n=0,1,2,…and the system of Hermite functions (8)hny=2nn!π-1/2e-y2/2Hny,n=0,1,2,…form an orthonormal basis for L2(R). By abuse of notation we also denote Hn to the one-dimensional space generated by hn(y) for n∈Z+. Further, define (9)Hn⊕=⨁k=0nHk.The one-dimensional projection P(n) from L2(R) onto Hn is given by (P(n)ϕ)(y)=〈ϕ,hn〉·hn(y). Thus, Pn=P(0)⊕⋯⊕P(n) is the orthogonal projection from L2(R) onto Hn⊕, and (10)Pnϕy=∑k=0nϕ,hk·hky=∑k=0nhky∫Rϕvhkvdv.On the other hand, we consider the unitary operatorU1:L2(C,e|z|2)→L2(R2,dxdy) defined by (U1φ)(z)=π-1e(1/2)zz¯φ(z) that transforms the space Fk2(C) into Fk(1), the set of all functions in L2(R2), which satisfy the following equation: (11)∂∂z¯+z2kf=0The image Fk(2) of the space Fk(1) under the unitary transform U2=I⊗F is the closure of the set of all smooth functions in L2(R2) which satisfy the equation (12)12k∂∂x-y+x-∂∂ykf=0,where F is the Fourier transform.Finally, we take the isomorphismU3:L2(R2)→L2(R2) defined by (13)U3fx,y=f12x+y,12x-yto transform the space Fk(2) onto the space Fk(3), which is the closure of the set of smooth functions satisfying the equation (14)12k/2∂∂y+ykf=0.In summary, the unitary operator U=U3U2U1 provides an isometric isomorphism from the space L2(C,e|z|2) into the space L2(R,dx)⊗L2(R,dy), under which the k poly-Fock space Fk2 is mapped into L2(R)⊗Hk⊕. We denote by B(n) and Bn the orthogonal projections from L2(C,e|z|2) onto F(n)2(C) and Fn2(C), respectively. The true k poly-Fock spaces F(k)2(C) are defined as follows: (15)F12C=F12C=F2C,k=1,Fk2C=Fk2C⊖Fk-12C,k>1.Thus, P(k) is true k-Bargmann projection from L2(C) into F(k)(C).The above construction is due to Vasilevski in [1], using the unitary operator U, they obtain the following characterizations:(1)
The true-poly-Fock spaceF(n)2(C) is mapped onto L2(R)⊗Hn-1.(2)
The true-poly-Fock projectionB(n) is unitary equivalent to the following one: (16)UBnU-1=I⊗Pn-1.(3)
The poly-Fock spaceFn2(C) is mapped onto L2(R)⊗Hn-1⊕.(4)
The poly-Fock projectionBn is unitary equivalent to the following one: (17)UBnU-1=I⊗Pn-1.We introduce the isometric embeddingR0,(n):L2(R)→L2(R2) by the rule (R0,(n)f)(x,y)=f(x)hn-1(y). Clearly, the adjoint operator R0,(n)∗:L2(R2)→L2(R+) is given by (18)R0,n∗φx=∫Rφx,vhn-1vdv.The previous operators satisfy the following relations: (19)R0,n∗R0,n=I:L2R⟶L2RR0,nR0,n∗=I⊗Pn-1:L2R⊗L2R⟶L2R⊗Hn-1.On the other hand, we introduce the operatorR(n)=R0,(n)∗U from L2(C,e|z|2) onto L2(R), and its restriction to F(n)2(C) is an isometric isomorphism. Thus, the adjoint operator R(n)∗=U∗R0,(n) is an isometric isomorphism from L2(R) onto the subspace F(n)2(C). Hence, these operators satisfy the following relations:(20)Rn∗Rn=Bn:L2C,ez2⟶Fn2CRnRn∗=I:L2R⟶L2R.Similarly, introduce the isometric embeddingR0,n:(L2(R))n→L2(R)⊗L2(R) by the rule (21)R0,nfx,y=∑k=1nfkxhk-1y=NnyfxT,where f=(f1,…,fn) and(22)Nny=h0y,…,hn-1yand the superscript T means that we are taking the transpose matrix.Further, the adjoint operatorR0,n∗:L2(R2)→(L2(R))n is given by (23)R0,n∗φx=∫Rφx,yh0ydy,…,∫Rφx,yhn-1ydy=∫Rφx,yNnydy.Since the image of R0,n is the space U(Fn2(C))=L2(R)⊗Hn-1⊕, hence these operators satisfy the following relations: (24)R0,n∗R0,n=I:L2Rn⟶L2Rn,R0,nR0,n∗=I⊗Pn-1:L2R2⟶L2R⊗Hn-1⊕.Now the operatorRn≔R0,n∗U from L2(C,e|z|2) onto (L2(R))n, and its restriction to Fn2(C) is an isometric isomorphism. Furthermore, the adjoint operator Rn∗=U∗R0,n is an isometric isomorphism from (L2(R))n onto the space Fn2(C). Hence, these operators satisfy the following relations:(25)Rn∗Rn=Bn:L2C,ez2⟶Fn2C,RnRn∗=I:L2Rn⟶L2Rn.


## 3. Toeplitz Operators with Horizontal Symbol

In this section we introduce a certain class of Toeplitz operators acting on the poly-Fock spaces, and we prove that they are unitarily equivalent to multiplication operators by continuous matrix-valued functions on(-∞,∞). Let a(z)=a(x) be a function in L∞(R) depending only on x=Rez and we called this function a horizontal symbol.Definition 1.
Leta be a function in L∞(R). The Toeplitz operator with symbol a acting on true-poly-Fock space (or poly-Fock space) is defined as (26)Ta:φ∈Fn2C⟼Bnaφ∈Fn2C.or(27)Ta:φ∈Fn2C⟼Bnaφ∈Fn2C,where B(n) and Bn are the orthogonal projections for true-poly-Fock space and poly-Fock space, respectively.In [2], K. Esmeral and N. Vasilevski show that every Toeplitz operator Ta with horizontal symbol a(x)∈L∞(R) acting on F2(C) is unitary equivalent to the multiplication operator γa(x)I=R0TaR0∗ acting on L2(R), where R0 is defined in Section 2. The function γa is given by(28)γax=π∫Ray+x2e-y2dy.The following theorem is a generalization of above result for Toeplitz operators with horizontal symbols acting on true-poly-Fock space.Theorem 2.
For anya(x)∈L∞(R), the Toeplitz operator T(n),a acting on F(n)2(C) is unitary equivalent to the multiplication operator γ(n),aI=R(n)T(n),aR(n)∗ acting on L2(R), where the function γ(n),a is given by(29)γn,ax=∫Rax+y2hn-1y2dy.Proof.
We know that the operatorR(n) is unitary and using (20), we obtain that the Toeplitz operator T(n),a is unitary equivalent to the following operators: (30)RnTn,aRn∗=RnBΠ,naBΠ,nRn∗=RnRn∗RnaRn∗RnRn∗=RnRn∗RnaRn∗RnRn∗=RnaRn∗=R0,n∗U3U2U1ayU1-1U2-1U3-1R0,n=R0,n∗U3ayU3-1R0,n=R0,n∗ax+y2R0,n.
Now calculate the explicit expression of the above operator(31)R0,n∗ax+y2R0,nfx=∫Rax+y2fxhn-1y2dy=γn,ax·fx,where f∈L2(R) and hn-1 is the Hermite function of degree n-1.We calledγ(n),a(x) the n spectral function for the Toeplitz operator with vertical symbol a in the true-poly-Fock space.Remark 3.
Notice that we obtain (28) from (29) taking n=1.The following result is an extension of above theorem for Toeplitz operators with horizontal symbols acting on poly-Fock space.Theorem 4.
For anya(x)∈L∞(R), the Toeplitz operator Tn,a acting on Fn2(C) is unitary equivalent to the matrix multiplication operator γn,a(x)I=RnTn,aRn∗ acting on (L2(R))n, where the matrix-valued function γn,a=(γijn,a) is given by(32)γn,ax≔∫Rax+y2NnyTNnydy.That is,(33)γijn,ax=∫Rax+y2hi-1yhj-1ydy,for i,j=1,…,n.Proof.
We have that the operatorRn is unitary and using (25), we obtain that the Toeplitz operator Tn,a is unitary equivalent to the following operators: (34)RnTn,aRn∗=RnBnaBnRn∗=RnRn∗RnaRn∗RnRn∗=RnRn∗RnaRn∗RnRn∗=RnaRn∗=R0,n∗U3U2U1ayU1-1U2-1U3-1R0,n=R0,n∗U3ayU3-1R0,n=R0,n∗ax+y2R0,n.
Now calculate the explicit expression of the above operator(35)R0,n∗ax+y2R0,nfx=R0,n∗ax+y2NnyTfx=∫Rax+y2NnyTfxNnydy=∫Rax+y2NnyNnyTfxdy=γn,axfx,where f=(f1,…,fn)∈(L2(R))n and Nn(y) is given by (22).
Therefore we obtain that each component ofγn,a is given by (33), which proves the theorem.Remark 5.
The component function (33) is equal to γijn,a(x)=a~∗(hihj), where ∗ denotes the convolution in R, a~(x)=a(x/2)∈L∞(R) and hihj∈L1(R). From [7][p.283, 32.45] it is guaranteed that the function γijn,a(x) belongs to Cb(R) where Cb(R) is the set of uniformly continuous functions in R.


## 4. Description of theC∗-Algebra Generated by Toeplitz Operators with Extended Horizontal Symbols

Denote byL∞{-∞,+∞}(R) the closed subspace of L∞(R) which consists of all functions having limit values at the “endpoints” -∞ and +∞; i.e., for each a∈L∞{-∞,∞}(R) the following limits exist (36)limy→-∞ay=a-andlimy→+∞ay=a+.We will identify the functions a∈L∞{0,∞}(R+) with their extensions a(z)=a(x) to the complex plane C, where y=Imz. We shall say that a∈L∞{0,∞}(R+) is an extended horizontal symbol.In this section we study theC∗-algebra generated by all the Toeplitz operators on Fn2(C) with extended horizontal symbols.Definition 6.
We define someC∗-algebras that we will use in this paper. (i)
Denote byG(n)H the set of horizontal spectral functions given by (37)GnH=γn,a:ax∈L∞-∞,+∞R.(ii)
Denote byGnH the set of horizontal spectral matrix-valued functions given by (38)GnH=γn,a:ax∈L∞-∞,+∞R.(iii)
Denote byT-∞,∞(n) the C∗-algebra generated by all the Toeplitz operators T(n),a acting on the true-poly-Fock space F(n)2(C), with a∈L∞{-∞,∞}(R).(iv)
Denote byT-∞,∞n the C∗-algebra generated by all the Toeplitz operators Tn,a acting on the poly-Fock space Fn2(C), with a∈L∞{-∞,∞}(R).Corollary 7.
TheC∗-algebra T-∞,∞(n) is isometrically isomorphic to the C∗-algebra G(n)H generated by G(n)H.Corollary 8.
TheC∗-algebra T-∞,∞n is isometrically isomorphic to the C∗-algebra GnH generated by GnH.The next lemma is important to describe the behavior at the infinity of the spectral matrix-valued-function related to Toeplitz operators with extended horizontal symbols on poly-Fock space.Lemma 9.
We consider a horizontal functiona(x)∈L∞{-∞,+∞}(R), and let (39)a-=limy→-∞axanda+=limy→+∞ax.Then the matrix-valued function γn,a(x) satisfies (40)a-I=limx→-∞γn,axanda+I=limx→+∞γn,ax.Proof.
First we consider the case whena-=0 (analogously we have a+=0). We proceed to show that the limit value at -∞ of each entry of γn,a is equal to zero. We know that hi-1hj-1∈L1(R); then (41)Cij=∫-∞∞hi-1yhj-1ydy<∞.Let ϵ>0 be a fixed number; hence there exists y0∈R+ such that (42)∫y0∞hi-1yhj-1ydy<ϵ.Moreover, there exist M>0 such that at<ϵ for t<-M. Under the above assumptions, we estimate the value of each entry of γn,a(x) as follows: (43)γijn,ax≤∫-∞y0ax+y2hi-1yhj-1ydy+∫y0∞ax+y2hi-1yhj-1ydy≤Cijmax-∞<y<y0ax+y2+a∞ϵ.If x<-y0-2M and y<y0, then (x+y)/2<-M. By the above, it is clear that (44)γijn,ax≤Cij+a∞ϵforx<-y0-2M,which implied the result for the limit value at -∞ when a-=0. By similar argument the result is valid for the limit value at ∞ when a+=0.
Now we consider the case whena-≠0. If b(x)=a(x)-a-, from the previous case we have that(45)limx→-∞γijn,ax=limx→-∞γijn,bx+limx→-∞γijn,a-x=a-∫-∞∞hi-1yhj-1ydy=a-δij.By similar argument, we obtain limx→+∞γijn,a(x)=a+δij. This completes the proof.Recall thatC0(R) is the set of continuous functions that vanishes at infinity and C(R¯) is the set of continuous functions belonging to L∞{-∞,+∞}(R).Remark 10.
Leta(x)∈L∞{-∞,+∞}(R) be an extended horizontal function; then (46)γijn,a∈C0Rifi≠j,γiin,a∈CR¯andγn,a∈CR¯,where γijn,a and γ(n),a are given by (29) and (32), for i,j=1,…,n, respectively. In particular, it is clear that γnnn,a=γ(n),a for each n∈N.We denote the algebra of alln×n matrices with complex entries by Mn(C) and we define the C∗-algebra C=Mn(C)⊗C(R¯) that consists of the algebra of all n×n matrices with entries in C(R¯). We introduce the algebra D, which is a C∗-subalgebra of C defined by (47)D=M∈C:M-∞,M∞∈CI.It is clear thatGnH is a C∗-subalgebra of D. We want to prove that GnH=D, where D is an algebra of type I. Thus, using a Stone-Weierstrass theorem [8], we just need to show that GnH separates all the pure states of D. Now, we proceed to describe the pure states of D.Notice thatD is a C∗-bundle and the fibers are given by (48)Dx0=MnC,x∈-∞,∞C,x=-∞orx=∞Moreover, the set of its pure states ofD is determined by the pure states on the fibers; i.e., each pure state of D has the form (49)fM=fx0Mx0,M∈D,where x0∈[-∞,∞], and fx0 is a pure state of D(x0); see [9] for more details.In [10], T. K. Lee characterizes the set of all states of the matrix algebra Mn(C). The author shows that each pure state of Mn(C) is given by a functional fv defined as(50)fwQ=Qv,vforQ∈MnCwhere v∈Sn={z∈Cn:z=1} and 〈·,·〉 denotes the usual inner product on Cn. Moreover, if v,w∈Sn such that fv=fw then v=tw, where t∈C and t=1.In consequence, we have that the set of pure states ofD consists of all functionals of the form (51)fx0,vM=Mx0v,v,M∈D,where x0∈[-∞,∞] and v∈Sn.In the case whenx0=-∞ or x0=∞, we just have one pure state which can be realized by fx0,v for each v∈Sn; i.e., (52)fx0,v=fx0,wforx0=-∞or∞,for all v,w∈Sn.Now, we will show thatGnH separates all the pure states of D. For this task we are going to use horizontal symbols of the form cβ(z)=χ(-∞,β](y), where χ(-∞,β] is the characteristic function of the set (-∞,β]⊂R. Thus for this function the spectral matrix-valued-function has the form(53)γn,cβx=∫Rcx+y2NnyNnyTdy=∫-∞2β-xNnyNnyTdy.To simplify the notation we made the following convention: γn,β=γn,cβ, where cβ(z)=χ(-∞,β](y) and β∈R¯.Letv∈Sn; we define the function x0∈R¯↦fx0,v(γn,β(x)), whose explicit form is given by (54)fx0,vγn,βx=γn,βx0v,v,=∫-∞2β-x0NnyNnyTdyv,v=∫-∞2β-x0NnyNnyTv,vdy=∫-∞2β-x0v,NnyNny,vdy=∫-∞2β-x0v,Nny2dy.In particular, from Lemma9 we have that f-∞,v(γn,β(x))=1 and f∞,v(γn,β(x))=0. Moreover, it is clear that 0<fx0,v(γn,β(x))<1 for all x0∈R. Hence we separated the unique pure state at x0=-∞ (or x0=∞) from every other pure state of the C∗-algebra D.Now we define the functionhv(y)=|〈v,Nn(y)〉|2 for y∈R. Notice that this function can be expressed as hv(y)=qv(y)e-y2, where(55)qvy=v0H0y+⋯+vn-1-1n-1Hn-1y2is a polynomial of degree at most 2n-2 nonnegative valued.The following lemma provides us a tool to show that theC∗-algebra GnH separates the pure states of D of the form fx0,v,fx1,w, where x0≠x1 with x0,x1∈R and v,w∈Sn.Lemma 11.
We assume thatv,w∈Sn, x0,x1∈R, and γn,β∈GnH, where β∈R. If we have that fx0,v(γn,β(x))=fx1,w(γn,β(x)) for all β∈R, then x0=x1 and |〈v,Nn(y)〉|2=|〈w,Nn(y)〉|2 for all y∈R.Proof.
By hypothesis we have thatfx0,v(γn,β(x))=fx1,w(γn,β(x)), which is equivalent to the following: (56)∫-∞2β-x0qvye-y2dy=∫-∞2β-x0qwye-y2dy,where qv is given by (55). Taking the derivative with respect to β of the above equation yields (57)qv2β-x0e-2β-x02=qw2β-x1e-2β-x12for all β∈R. We can rewrite the above equation as (58)qv2β-x0=qw2β-x1e-2β-x12+2β-x02=qw2β-x1e22βx1-x0+x02-x12We know that qv,qw are polynomials, which implies that the above equation is valid if and only if the exponential part is constant with respect to β. Hence we obtain that x0=x1; using this fact it is clear that qv(2β-x0)=qw(2β-x0) for all β∈R. Therefore we have that x0=x1 and qv=qw which is equivalent to |〈v,Nn(y)〉|2=|〈w,Nn(y)〉|2 for all y∈R.Remark 12.
A consequence of the above lemma is that ifx0≠x1 and v,w∈Sn, then there exist β0∈R such that fx0,v(γn,β0(x))≠fx1,w(γn,β0(x)); i.e., the spectral matrix-valued-function γn,β0 separated the pure states fx0,v and fx1,w.Lety1,⋯,yn be real numbers different from each other and recall that Nn(y)=e-y2(H0(y),H1(y),…,Hn-1(y)). We define the matrix N where the row k is equal to Nn(yk). Thus (59)N=DH0y1H1y1⋯Hn-1y1H0y2H1y2⋯Hn-1y2⋮⋮⋮H0ynH1yn⋯Hn-1yn,where D is the diagonal matrix given by diag{e-y12/2,⋯,e-yn2/2}. Notice that 2k/2/πk! is the leading coefficient of Hk(y); thus (60)Hky=2k/2πk!yk+lowerdegreetermsNow, we calculate the determinant ofN using the properties of multilineality, alternativity, and Vandermonde’s formula; we have the following relations:(61)detN=2n-1n/4·detDπn/2∏k=0n-1k!1y1+a0y12+b1y1+b0⋯y1n-1+⋯1y2+a0y22+b1y2+b0⋯y2n-1+⋯⋮⋮⋮⋮1yn+a0yn2+b1yn+b0⋯ynn-1+⋯=2n-1n/4·e-y12+⋯+yn2πn/2∏k=0n-1k!1y1y12⋯y1n-11y2y22⋯y2n-1⋮⋮⋮⋮1ynyn2⋯ynn-1=2n-1n/4·e-y12+⋯+yn2πn/2∏k=0n-1k!∏1≤i<j≤nyj-yi≠0.To complete the proof of the fact that theC∗-algebra GnH separates all the pure states of D, only missing step is to separate the pure states of the forms fx0,w and fx0,v, where v,w∈Sn and x0∈R.Lemma 13.
Givenv,w∈Sn and x0∈R being fixed, consider the spectral matrix-valued-functions γn,α(x),γn,β(x)∈GnH for all α,β∈R, which are given by (32). If fx0,v(γn,α(x)γn,β(x))=fx0,w(γn,α(x)γn,β(x)) for all α,β∈R, then v=tw, where t∈C and |t|=1.Proof.
From Lemma11, we have |〈v,Nn(y)〉|2=|〈w,Nn(y)〉|2 for all y, which implies that there exist a function θ:R→R such that(62)v,Nny=eiθyw,Nnyforally∈R.
For eachu∈Sn, we define the function Hu:R2→C given by (63)Huα,β=fx0,vγn,αxγn,βx=∫-∞2α-x0u¯TNnyNnyTdy·∫-∞2β-x0NntNntTudt.Without loss of generality we can assume that x0=0, just to simplify the calculations. Thus we calculate the second derivative of Hu with respect to α and β; we obtain(64)∂2Hu∂α∂βα2,β2=u¯TNnαNnαTNnβNnβTu(65)=Nnα,uNnα,Nnβu,Nnβ
By hypothesis, we have thatHv=Hw which implies that ∂2Hv/∂α∂β=∂2Hw/∂α∂β; using this fact and (62) and (64) we obtain (66)Nnα,wNnα,Nnβw,Nnβ=eiθβ-θαNnα,wNnα,Nnβw,Nnβfor all α,β∈R.
It is clear that there existβ0∈R fixed such that 〈w,Nn(β0)〉≠0. Thus, turn out the above equation as follows: (67)1-eiθβ0-θαNnα,wNnα,Nnβ0=0for all α∈R.
Notice that〈Nn(α),w〉〈Nn(α),Nn(β0)〉 is nonzero polynomial with respect to α; thus the above equation implies that the function θ is constant. From (62), we obtain that 〈v-eiθ0w,Nn(y)〉=0 for all y∈R. Using (61) it is clear that v-eiθ0=0, which implies the result.Remark 14.
The above result completes the proof that the algebraGnH separates to all pure states of D.The noncommutative Stone-Weierstrass conjecture: let B 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.This conjecture for aC∗-algebra type I was proved by I. Kaplansky in [8]. In consequence, we have proved that the algebra GnH is equal to D. From Corollary 8 we have that the algebra of Toeplitz operators T-∞,∞n is isometric and isomorphic to algebra D. In summary, we have the following result.Theorem 15.
TheC∗-algebra T-∞,∞n is isomorphic and isometric to the C∗-algebra D. The isomorphism is given by (68)T-∞,∞n:Tn,a⟼SymTn,ax=γn,ax,where γn,a(x) is given in (32).Corollary 16.
TheC∗-algebra T-∞,∞(n) is isomorphic and isometric to the commutative C∗-algebra C[-∞,∞]. The isomorphism is given by (69)T-∞,∞n:Tn,a⟼SymTn,ax=γn,ax,where γ(n),a(x) is given in (29).



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*Source: 8031259-2018-07-16.xml*