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

1. Introduction

Recall that the n poly-Fock space is denoted by [mathematical expression not reproducible] and consists of the n-analytic functions which satisfy the equation

[mathematical expression not reproducible] (1)

The n true poly-Fock space is denote by [F.sup.s.sub.(n)] (C), which consists of the all true-n-analytic functions; i.e.,

[mathematical expression not reproducible], (2)

for n [greater than or equal to] 1, and [F.sup.2.sub.(0)] (C) = {0}.

It is clear that [F.sup.2.sub.1] (C) is the classical Fock space on the complex plane C, which is also denoted by [F.sup.2](C).

In [1], N. Vasilevski proved that [L.sup.2](C) has a decomposition as a direct sum of the n true poly-Fock and n true antipoly-Fock spaces:

[mathematical expression not reproducible] (3)

Moreover, they proved that the spaces [F.sup.2.sub.(n)](C) are isomorphic and isometric to [L.sub.2](R) [R] [cross product] [H.sub.n-1], where [H.sub.n-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 socalled 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.sub.[infinity]] - 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], Sanchez-Nungaray and Vasilevski studied Toeplitz operators with pseudodifferential symbols acting on poly-Bergman spaces upper half plane. A different approach by Hutnlk, Maximenko, and Miskova 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 polyBergman 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. Ramirez-Ortega and A. Sanchez-Nungaray described the C* -algebra generated by the Toeplitz operators with bounded vertical symbols and acting over each polyBergman space in the upper plane [A.sup.2.sub.n]([PI]). They considered bounded vertical symbols that have limit values at y = 0, [infinity] and prove that the C* -algebra generated by the Toeplitz operator acting on [A.sup.2.sub.n] ([PI]) with this kind of symbols is isomorphic and isometric to the C*-algebra of matrix-valued functions of the compact [0, [infinity]]. Similar result can be found in [6], where M. Loaiza and J. Ramirez-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 classified C*-algebra generated by the Toeplitz operators with bounded vertical symbols with limits at -[infinity] and [infinity] acting over poly-Fock space in the complex plane.

This paper is organized as follows. In Section 2 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 [[gamma].sup.n,[alpha]] (x)I acting on ([L.sub.2]([R.sub.+])).sup.n, where [[gamma].sup.n,a] (x) is a continuous matrix-valued function on (-[infinity], [infinity]). Finally, in Section 4, we describe the pure states of the algebra

[mathematical expression not reproducible] (4)

We prove that the C* algebra [T.sup.(n).sub.[[infinity],[infinity]] generated by Toeplitz operator with bounded vertical symbols that have limit values at y = -[infinity], [infinity] 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 [member of] C, with the usual complex conjugation [mathematical expression not reproducible] . The Gaussian measure on C is given by

[mathematical expression not reproducible] (5)

where dv(z) = dx dy is the usual Euclidean measure on [R.sup.2] = C.

The Hilbert space [mathematical expression not reproducible] square integrable functions on C with the inner product

[mathematical expression not reproducible]. (6)

The closed subspace [mathematical expression not reproducible] consisting of all analytic functions is called the Fock or Segal-Bargmann space. Also, the Fock space [F.sup.2](C) can be defined as the closure of the set of all smooth functions satisfying the equation [bar.[partial derivative]][phi] = 0. Similarly, given a natural number k, the k poly-Fock space [F.sup.2.sub.k] (C) is the closure of the set of all smooth functions in [mathematical expression not reproducible].

Recall that the Hermite polynomial [H.sub.n](y) of degree n is defined by

[mathematical expression not reproducible] (7)

and the system of Hermite functions

[mathematical expression not reproducible] (8)

form an orthonormal basis for [L.sub.2](R). By abuse of notation we also denote [H.sub.n] to the one-dimensional space generated by [h.sub.n](y) for n [member of] [Z.sub.+]. Further, define

[mathematical expression not reproducible]. (9)

The one-dimensional projection from [P.sub.(n) from [L.sub.2]] [L.sub.2](R) onto [H.sub.n] is given by [mathematical expression not reproducible] is the orthogonal projection from [L.sub.2](R) onto [H.sup.[cross product].sub.n], and

[mathematical expression not reproducible] (10)

On the other hand, we consider the unitary operator [mathematical expression not reproducible] defined by ([mathematical expression not reproducible] that transforms the space [F.sup.2.sub.k] (C) into [F.sup.(1).sub.k], the set of all functions in [L.sub.2]([R.sup.2]), which satisfy the following equation:

[mathematical expression not reproducible] (11)

The image [F.sup.(2).sub.k] of the space [F.sup.(1).sub.k] under the unitary transform [U.sub.2] = I [cross product] F is the closure of the set of all smooth functions in [L.sub.2]([R.sup.2]) which satisfy the equation

[mathematical expression not reproducible] (12)

where F is the Fourier transform.

Finally, we take the isomorphism [U.sub.3]: [L.sub.2] ([R.sup.2]) [right arrow] [L.sub.2]([R.sup.2]) defined by

[mathematical expression not reproducible] (13)

to transform the space [F.sup. (2).sub.k] onto the space [F.sup.(3).sub.k], which is the closure of the set of smooth functions satisfying the equation

[mathematical expression not reproducible] (14)

In summary, the unitary operator U = [U.sub.3] [U.sub.2][U.sub.1] provides an isometric isomorphism from the space [mathematical expression not reproducible] into the space [L.sub.2](R,dx) [cross product] [L.sub.2](R,dy), under which the k poly-Fock space [F.sup.2.sub.k] is mapped into [L.sub.2](R) [cross product] [H.sup.[cross product].sub.k] . We denote by [B.sub.(n)] and [B.sub.n] the orthogonal projections from [mathematical expression not reproducible], respectively The true k poly-Fock spaces [F.sup.2.sun.(k)](C) are defined as follows:

[mathematical expression not reproducible] (15)

Thus, P(k) is true k-Bargmann projection from [L.sub.2](C) into [F.sub.(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 space [F.sup.2.sub.(n)] (C) is mapped onto [L.sub.2](R) [cross product] [H.sub.n-1].

(2) The true-poly-Fock projection [B.sub.(n)] is unitary equivalent to the following one:

U[B.sub.(n) [U.sup.-1] = I [cross product] [P.sub.(n-1)]. (16)

(3) The poly-Fock space [F.sup.2.sub.n](C) is mapped onto [L.sub.2](R) [cross product] [H.sup.[direct sum].sub.n-1].

(4) The poly-Fock projection [B.sub.n] is unitary equivalent to the following one:

[UB.sub.n][U.sup.-1] = I [cross product] [P.sub.n-1]. (17)

We introduce the isometric embedding [R.sub.0,(n]): [L.sub.2] (R) [right arrow] [L.sub.2]([R.sup.2]) by the rule [mathematical expression not reproducible]. Clearly, the adjoint operator [R.sup.*.sub.0,(n)] : [L.sub.2]([R.sup.2]) [right arrow] [L.sub.2]([R.sup.+]) is given by

[mathematical expression not reproducible]. (18)

The previous operators satisfy the following relations:

[mathematical expression not reproducible] (19)

On the other hand, we introduce the operator [mathematical expression not reproducible], onto [L.sub.2](R), and its restriction to [F.sup.2.sub.(n)] (C) is an isometric isomorphism. Thus, the adjoint operator [R.sup.*.sub.(n)] = [U.sup.*] [R.sub.0,(n)] is an isometric isomorphism from [L.sub.2](R) onto the subspace [F.sup.2.sub.(n)]) (C). Hence, these operators satisfy the following relations:

[mathematical expression not reproducible] (20)

Similarly, introduce the isometric embedding [R.sub.0,n] ([L.sub.2][(R)).sup.n] [right arrow] [L.sub.2](R) 8i2(R) by the rule

[mathematical expression not reproducible] (21)

where f = ([f.sub.1],..., [f.sub.n]) and

[N.sub.n] (y) = ([h.sub.0] (y),..., [h.sub.n-1] (y)) (22)

and the superscript T means that we are taking the transpose matrix.

Further, the adjoint operator [mathematical expression not reproducible]

is given by

[mathematical expression not reproducible] (23)

Since the image of [R.sub.0,n] is the space [mathematical expression not reproducible], hence these operators satisfy the following relations:

[mathematical expression not reproducible] (24)

Now the operator [mathematical expression not reproducible] onto [([L.sub.2](R)).sup.n], and its restriction to [F.sup.2.sub.n] (C) is an isometric isomorphism. Furthermore, the adjoint operator [R.sup.*.sub.n] = U* [R.sub.0,n] is an isometric isomorphism from [([L.sup.2](R)).sup.n] onto the space [F.sup.2.sub.n] (C). Hence, these operators satisfy the following relations:

[mathematical expression not reproducible] (25)

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 (-[infinity], [infinity]). Let a(z) = a(x) be a function in LTO(R) depending only on x = Re z and we called this function a horizontal symbol.

Definition 1. Let a be a function in [L.sub.[infinity]] (R). The Toeplitz operator with symbol a acting on true-poly-Fock space (or poly-Fockspace) is defined as

[mathematical expression not reproducible]. (26)

or

[mathematical expression not reproducible], (27)

where [B.sub.(n)] and [B.sub.n] are the orthogonal projections for truepoly-Fock space and poly-Fock space, respectively.

In [2], K. Esmeral and N. Vasilevski show that every Toeplitz operator [T.sub.a] with horizontal symbol a(x) [member of] [L.sub.[infinity]] (R) acting on [F.sup.2](C) is unitary equivalent to the multiplication operator [[gamma].sub.a](x)I = [R.sub.0] [T.sub.a] [R.sup.*.sub.0] acting on [L.sub.2](R), where [R.sub.0] is defined in Section 2. The function [[gamma].sub.a] is given by

[mathematical expression not reproducible]. (28)

The following theorem is a generalization of above result for Toeplitz operators with horizontal symbols acting on truepoly-Fock space.

Theorem 2. For any a(x) [member of] [L.sub.[infinity]] (R), the Toeplitz operator [T.sub.(n),a] acting on [F.sup.2.sub.(n)] (C) is unitary equivalent to the multiplication operator [mathematical expression not reproducible] acting on [L.sub.2](R), where the function [[gamma].sub.(n),a] is given by

[mathematical expression not reproducible]. (29)

Proof. We know that the operator [R.sub.(n)] is unitary and using (20), we obtain that the Toeplitz operator [T.sub.(n),a] is unitary equivalent to the following operators:

[mathematical expression not reproducible] (30)

Now calculate the explicit expression of the above operator

[mathematical expression not reproducible] (31)

where f [member of] [L.sub.2](R) and [h.sub.n-1] is the Hermite function of degree n-1.

We called [gamma].sub.(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 polyFock space.

Theorem 4. For any a(x) [member of] [L.sub.[infinity]] (R), the Toeplitz operator [T.sub.n,a] acting on [F.sup.2.sub.n] (C) is unitary equivalent to the matrix multiplication operator [mathematical expression not reproducible], where the matrix-valued function [[gamma].sup.n,a] = ([[gamma].sup.n,a].sub.ij]is given by

[mathematical expression not reproducible] (32)

That is,

[mathematical expression not reproducible] (33)

for i, j = 1,..., n.

Proof. We have that the operator Rn is unitary and using (25), we obtain that the Toeplitz operator [T.sub.n,a] is unitary equivalent to the following operators:

[mathematical expression not reproducible] (34)

Now calculate the explicit expression of the above operator

[mathematical expression not reproducible] (35)

where f = ([f.sub.1],..., [f.sub.n]) [member of] [([L.sub.2](R)).sup.2] and [N.sub.n](y) is given by (22).

Therefore we obtain that each component of [[gamma].sup.n,a] is given by (33), which proves the theorem.

Remark 5. The component function (33) is equal to [mathematical expression not reproducible], where * denotes the convolution in [mathematical expression not reproducible] it is guaranteed that the function [[gamma].sup.n,a].sub.ij] (x) belongs to [C.sub.b](R) where [C.sub.b](R) is the set of uniformly continuous functions in R.

4. Description of the [C.sup.*]-Algebra Generated by Toeplitz Operators with Extended Horizontal Symbols

Denote by [L.sup.{-[infinity],+[infinity]}.sub.[infinity]](R) the closed subspace of [L.sub.[infinity]](R) which consists of all functions having limit values at the "endpoints" -[infinity] and +[infinity]; i.e., for each a [member of] [L.sup.{-[infinity],+[infinity]}.sub.[infinity]] (R) the following limits exist

[mathematical expression not reproducible] (36)

We will identify the functions a [member of] [L.sup.{-[infinity],+[infinity]}.sub.[infinity]] ([R.sub.+]) with their extensions a(z) = a(x) to the complex plane C, where y = Im z. We shall say that a [member of] [L.sup.{-[infinity],+[infinity]}.sub.[infinity]](R+) is an extended horizontal symbol.

In this section we study the [C.sup.*]-algebra generated by all the Toeplitz operators on [F.sup.2.sub.n] (C) with extended horizontal symbols.

Definition 6. We define some C* -algebras that we will use in this paper.

(i) Denote by [g.sup.H.sub.(n)] the set of horizontal spectral functions given by

[mathematical expression not reproducible]. (37)

(ii) Denote by [g.sup.H.sub.(n)] the set of horizontal spectral matrix-valued functions given by

[mathematical expression not reproducible]. (38)

(iii) Denote by [T.sup.(n).sub.-[infinity],[infinity]] [degrees] the C* -algebra generated by all the Toeplitz operators [T.sub.(n)a] acting on the true-poly-Fock space [F.sup.2.sub.(n)](C), with a [member of] [L.sup.{-[infinity],+[infinity]}.sub.[infinity]] (R).

(iv) Denote by [T.sup.(n).sub.-[infinity],[infinity]] the [C.sup.*] -algebra generated by all the Toeplitz operators [T.sub.n,a] acting on the poly-Fock space [F.sup.2.sub.n](C), with a [member of] [L.sup.{-[infinity],+[infinity]}.sub.[infinity]] (R).

Corollary 7. The C*-algebra [T.sup.(n).sub.-[infinity],[infinity]] is isometrically isomorphic to the C*-algebra [K.sup.H.sub.n] generated by [G.sup.H.sub.n].

Corollary 8. The C*-algebra [T.sup.(n).sub.-[infinity],[infinity]] is isometrically isomorphic to the C*-algebra [K.sup.H.sub.n] generated by [G.sup.H.sub.n].

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 polyFock space.

Lemma 9. We consider a horizontal function a(x) [member of] [L.sup.{-[infinity],+[infinity]}.sub.[infinity]] and let

[mathematical expression not reproducible] (39)

Then the matrix-valued function [gamma].sup.n,a](x) satisfies

[mathematical expression not reproducible] (40)

Proof. First we consider the case when [a.sub.-] = 0 (analogously we have [a.sub.+] = 0). We proceed to show that the limit value at -[infinity] of each entry of [[gamma].sup.n,a] is equal to zero. We know that [h.sub.i-1] [h.sub.j-1] [member of] [L.sub.1](R);then

[mathematical expression not reproducible] (41)

Let [epsilon] >0 be a fixed number; hence there exists [y.sub.0] [member of] [R.sub.+] such that

[mathematical expression not reproducible] (42)

Moreover, there exist M > 0 such that [absolute value of a(t)] < [member of] for t < -M. Under the above assumptions, we estimate the value of each entry of [[gamma].sup.n,a] (x) as follows:

[mathematical expression not reproducible] (43)

If x < - [y.sub.0] - [square root of 2M] and y < [y.sub.0], then (x + y)/ [square root of < -M. By the above, it is clear that

[mathematical expression not reproducible], (44)

which implied the result for the limit value at -[infinity] when a- = 0. By similar argument the result is valid for the limit value at [infinity] when [a.sub.+] = 0.

Now we consider the case when a- [not equal to] 0. If b(x) = a(x)-a-, from the previous case we have that

[mathematical expression not reproducible] (45)

By similar argument, we obtain [mathematical expression not reproducible]. This completes the proof.

Recall that [C.sub.0](R) is the set of continuous functions that vanishes at infinity and C([bar.R]) is the set of continuous functions belonging to [L.sup.{-[infinity],+[infinity]}.sub.[infinity]] (R).

Remark 10. Let a(x) [member of] [L.sup.{-[infinity],+[infinity]}.sub.[infinity]] (R) be an extended horizontal function; then

[mathematical expression not reproducible] (46)

where [[gamma].sup.n,a.sub.i,j] and [[gamma].sub.(n),a] are given by (29) and (32), for i, j = 1,, n, respectively. In particular, it is clear that [[gamma].sup.n,a.sub.n,n] and [[gamma].sub.(n),a] for each n [member of] N.

We denote the algebra of all nxn matrices with complex entries by [M.sub.n](C) and we define the C*-algebra C = [M.sub.n](C)[cross product] C([bar.R]) that consists of the algebra of all nxn matrices with entries in C([bar.R]). We introduce the algebra S, which is a C*subalgebra of C defined by

D = {M [member of] C : M(-[infinity]), M([infinity]) [member of] CO}. (47)

It is clear that [C.sup.H.sub.n] is a C*-subalgebra of D. We want to prove that [C.sup.H.sub.n] = D, where D is an algebra of type I. Thus, using a Stone-Weierstrass theorem [8], we just need to show that [C.sup.H.sub.n] separates all the pure states of D. Now, we proceed to describe the pure states of D.

Notice that D is a C*-bundle and the fibers are given by

[mathematical expression not reproducible] (48)

Moreover, the set of its pure states of D is determined by the pure states on the fibers; i.e., each pure state of D has the form

[mathematical expression not reproducible], (49)

where [mathematical expression not reproducible] is a pure state of D([x.sub.0]); see [9] for more details.

In [10], T. K. Lee characterizes the set of all states of the matrix algebra [M.sub.n](C). The author shows that each pure state of [M.sub.n](C) is given by a functional [f.sub.v] defined as

[f.sub.w] (Q) = (Qv, v) for Q [member of] [M.sub.n] (C) (50)

where [mathematical expression not reproducible] denotes the usual inner product on [C.sup.n]. Moreover, if v, w [member of] [S.sup.n] such that [f.sub.v] = [f.sub.w] then v = tw, where t [member of] C and [absolute value of t] = 1.

In consequence, we have that the set of pure states of D consists of all functionals of the form

[mathematical expression not reproducible], (51)

where [x.sub.0] [member of] [-[infinity], [infinity]] and v [member of] [S.sup.n].

In the case when [x.sub.0] = -[infinity] or [x.sub.0] = [infinity], we just have one pure state which can be realized by [mathematical expression not reproducible] for each v [member of] [S.sup.n]; i.e.,

[mathematical expression not reproducible](52)

for all v, w [member of] [S.sup.n].

Now, we will show that [C.sup.H.sub.n] separates all the pure states of D. For this task we are going to use horizontal symbols of the form [mathematical expression not reproducible] is the characteristic function of the set (-[infinity], [beta]] [subset] R. Thus for this function the spectral matrix-valued-function has the form

[mathematical expression not reproducible] (53)

To simplify the notation we made the following convention: [mathematical expression not reproducible].

Let v [member of] [S.sup.n]; we define the function [mathematical expression not reproducible], whose explicit form is given by

[mathematical expression not reproducible] (54)

In particular, from Lemma 9 we have that [mathematical expression not reproducible]. Moreover, it is clear that 0 < [mathematical expression not reproducible]. Hence we separated the unique pure state at [x.sub.0] = -[infinity] (or [x.sub.0] = [infinity]) from every other pure state of the C*-algebra D.

Now we define the function [h.sub.v](y) = [absolute value of (v, [N.sub.n](y))].sup.2] for y [member of] R. Notice that this function can be expressed as [mathematical expression not reproducible], where

[mathematical expression not reproducible] (55)

is a polynomial of degree at most 2n- 2 nonnegative valued.

The following lemma provides us a tool to show that the C*-algebra [C.sup.H.sub.n] separates the pure states of D of the form [mathematical expression not reproducible], where [x.sub.0] [not equal to] [x.sub.1] with [x.sub.0], [x.sub.1] [member of] R and v,w [member of] Sn.

Lemma 11. We assume that [mathematical expression not reproducible].

Proof. By hypothesis we have that [mathematical expression not reproducible], which is equivalent to the following:

[mathematical expression not reproducible] (56)

where [q.sub.v] is given by (55). Taking the derivative with respect to [beta] of the above equation yields

[mathematical expression not reproducible] (57)

for all [beta] [member of] R. We can rewrite the above equation as

[mathematical expression not reproducible] (58)

We know that [q.sub.v], [q.sub.w] are polynomials, which implies that the above equation is valid if and only if the exponential part is constant with respect to [beta]. Hence we obtain that [x.sub.0] = [x.sub.1]; using this fact it is clear that [mathematical expression not reproducible] for all [beta] [member of] R. Therefore we have that [x.sub.0] = [x.sub.1] and [q.sub.V] = [q.sub.w] which is equivalent to [absolute value of <v, [N.sub.n](y))].sup.2] = [absolute value of <w, [N.sub.n](y))].sup.2] for all y [member of] R.

Remark 12. A consequence of the above lemma is that if [x.sub.0] [not equal to] [x.sub.1] and v, w [member of] [S.sup.n], then there exist [[beta].sub.0] [member of] R such that [mathematical expression not reproducible]; i.e., the spectral matrix-valued-function [mathematical expression not reproducible] separated the pure states [mathematical expression not reproducible].

Let [y.sub.1],..., [y.sub.n] be real numbers different from each other and recall that [mathematical expression not reproducible]. We define the matrix N where the row k is equal to [N.sub.n]([y.sub.k]). Thus

[mathematical expression not reproducible] (59)

where D is the diagonal matrix given by [mathematical expression not reproducible]. Notice that [2.sup.k/2]/[square root of [pi]k!] is the leading coefficient of [H.sub.k](y); thus

[mathematical expression not reproducible] (60)

Now, we calculate the determinant of N using the properties of multilineality, alternativity, and Vandermonde's formula; we have the following relations:

[mathematical expression not reproducible] (61)

To complete the proof of the fact that the C* -algebra [C.sup.H.sub.n] separates all the pure states of D, only missing step is to separate the pure states of the forms [mathematical expression not reproducible], where v, w [member of] [S.sup.n] and [x.sub.0] [member of] R.

Lemma 13. Given v, w [member of] [S.sup.n] and [x.sub.0] [member of] R being fixed, consider the spectral matrix-valued-functions [mathematical expression not reproducible] for all [alpha], [beta] R, which are given by (32). If [mathematical expression not reproducible] for all [alpha], [beta] [member of] R, then v = tw, where t [member of] C and [absolute value of t] = 1.

Proof. From Lemma 11, we have [absolute value of (v,[N.sub.n](y))].sup.2] = [absolute value of (w,[N.sub.n](y))].sup.2] for all y, which implies that there exist a function [theta] : R [right arrow] R such that

[mathematical expression not reproducible] (62)

For each u [member of] [S.sup.n], we define the function [H.sub.u]: [R.sup.2] [right arrow] C given by

[mathematical expression not reproducible] (63)

Without loss of generality we can assume that [x.sub.0] = 0, just to simplify the calculations. Thus we calculate the second derivative of [H.sub.u] with respect to [alpha] and [beta]; we obtain

[mathematical expression not reproducible] (64)

[mathematical expression not reproducible] (65)

By hypothesis, we have that [H.sub.V] = [H.sub.w] which implies that [mathematical expression not reproducible]; using this fact and (62) and (64) we obtain

[mathematical expression not reproducible] (66)

for all [alpha], [beta] [member of] R.

It is clear that there exist [[beta].sub.0] [member of] R fixed such that (w, [N.sub.n] ([[beta].sub.0])) = 0. Thus, turn out the above equation as follows:

[mathematical expression not reproducible] (67)

for all [alpha] [member of] R.

Notice that ([N.sub.n]([alpha]),w)([N.sub.n]([alpha]), [N.sub.n]([[beta].sub.0])) is nonzero polynomial with respect to [alpha]; thus the above equation implies that the function [theta] is constant. From (62), we obtain that [mathematical expression not reproducible] for all y [member of] R. Using (61) it is clear that [mathematical expression not reproducible], which implies the result.

Remark 14. The above result completes the proof that the algebra [C.sup.H.sub.n] separates to all pure states of D.

The noncommutative Stone-Weierstrass conjecture: let B be a [C.sup.*]-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 a C*-algebra type I was proved by I. Kaplansky in [8]. In consequence, we have proved that the algebra [C.sup.H.sub.n] is equal to D. From Corollary 8 we have that the algebra of Toeplitz operators [T.sup.n.sub.-[infinity],[infinity]] is isometric and isomorphic to algebra D. In summary, we have the following result.

Theorem 15. The C*-algebra [T.sup.(n).sub.-[infinity],[infinity]]is isomorphic and isometric to the C*-algebra D. The isomorphism is given by

[mathematical expression not reproducible], (68)

where [[gamma].sup.n,a] (x) is given in (32).

Corollary 16. The C*-algebra [T.sup.(n).sub.-[infinity],[infinity]] is isomorphic and isometric to the commutative C*-algebra C[-[infinity],[infinity]]. The isomorphism is given by

[mathematical expression not reproducible], (69)

where [gamma].sub.(n) a](x) isgiven in (29).

https://doi.org/10.1155/2018/8031259

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was partially supported by the Conacyt Project, Mexico.

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Armando Sanchez-Nungaray (iD), (1) Carlos Gonzalez-Flores, (2) Raquiel Rufino Lopez-Martinez, (1) and Jorge Luis Arroyo-Neri (1)

(2) Instituto Politecnico Nacional, Escuela Superior de Ingenieria Mecanica y Electrica, Unidad Profesional Adolfo Lopez Mateos, Av. Instituto Politecnico Nacional s/n, Col. Zacatenco, Del. Gustavo A. Madero, 07738 Ciudad de Mexico, Mexico

Correspondence should be addressed to Armando Sanchez-Nungaray; armsanchez@uv.mx

Received 12 April 2018; Accepted 13 June 2018; Published 16 July 2018