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

We describe the C∗-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 = Re z and have limit values at y = −∞ and y = ∞.TheC∗ algebra generated with this kind of symbols is isomorphic to theC∗-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.


Introduction
Recall that the  poly-Fock space is denoted by  2  (C) ⊂  2 (C,  −|| 2  ) and consists of the -analytic functions which satisfy the equation The  true poly-Fock space is denote by  2 () (C), which consists of the all true--analytic functions; i.e., for  ≥ 1, and  2 (0) (C) = {0}.It is clear that  2  1 (C) is the classical Fock space on the complex plane C, which is also denoted by  2 (C).
In [1], N. Vasilevski proved that  2 (C) has a decomposition as a direct sum of the  true poly-Fock and  true antipoly-Fock spaces: Moreover, they proved that the spaces  2 () (C) are isomorphic and isometric to  2 (R) ⊗  −1 , where  −1 is the onedimensional space generated by Hermite function of order  − 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  * -algebra generated by them.They showed that any Toeplitz operator with  ∞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  * -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  * -algebra generated by the Toeplitz operators with bounded vertical symbols and acting over each poly-Bergman space in the upper plane A 2  (Π).They considered bounded vertical symbols that have limit values at  = 0, ∞ and prove that the  * -algebra generated by the Toeplitz operator acting on A 2  (Π) with this kind of symbols is isomorphic and isometric to the  * -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  * -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  * -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 Section 2 we introduce preliminary results about the -polyanalytic function spaces and their relationship with the Hermite polynomials.In Section 3 we prove that every Toeplitz operator with bounded horizontal symbol () acting on Fock space is unitary equivalent to a multiplication operator  , () acting on ( 2 (R + ))  , where  , () is a continuous matrix-valued function on (−∞, ∞).Finally, in Section 4, we describe the pure states of the algebra We prove that the  * algebra T () −∞,∞ generated by Toeplitz operator with bounded vertical symbols that have limit values at  = −∞,∞ acting on Fock space is isomorphic and isometric to the  * -algebra D.

Poly-Fock Space on the Complex Plane
In this work we use the following standard notation:  =  +  ∈ C, with the usual complex conjugation ; thus || 2 = ⋅.The Gaussian measure on C is given by where V() =   is the usual Euclidean measure on R The closed subspace  2 (C) of  2 (C,  || 2 ) consisting of all analytic functions is called the Fock or Segal-Bargmann space.Also, the Fock space  2 (C) can be defined as the closure of the set of all smooth functions satisfying the equation  = 0. Similarly, given a natural number , the  poly-Fock space  2  (C) is the closure of the set of all smooth functions in  2 (C,  || 2  ) satisfying    = 0.
Recall that the Hermite polynomial   () of degree  is defined by and the system of Hermite functions form an orthonormal basis for  2 (R).By abuse of notation we also denote   to the one-dimensional space generated by ℎ  () for  ∈ Z + .Further, define The one-dimensional projection is the orthogonal projection from  2 (R) onto  ⊕  , and On the other hand, we consider the unitary operator , the set of all functions in  2 (R 2 ), which satisfy the following equation: The image  (2)   of the space  (1)   under the unitary transform  2 =  ⊗  is the closure of the set of all smooth functions in  2 (R 2 ) which satisfy the equation where  is the Fourier transform.Finally, we take the isomorphism to transform the space  (2)   onto the space  (3)  , which is the closure of the set of smooth functions satisfying the equation In summary, the unitary operator  =  3  2  1 provides an isometric isomorphism from the space  2 (C,  || 2 ) into the space  2 (R, ) ⊗  2 (R, ), under which the  poly-Fock space  2   is mapped into  2 (R) ⊗  ⊕  .We denote by  () and   the orthogonal projections from  2 (C,  || 2 ) onto  2 () (C) and  2  (C), respectively.The true  poly-Fock spaces  2 () (C) are defined as follows: (15) The above construction is due to Vasilevski in [1], using the unitary operator , they obtain the following characterizations: (1) The true-poly-Fock space (2) The true-poly-Fock projection  () is unitary equivalent to the following one: (3) The poly-Fock space  2  (C) is mapped onto  2 (R) ⊗  ⊕ −1 .(4) The poly-Fock projection   is unitary equivalent to the following one: We introduce the isometric embedding  0,() :  2 (R) →  2 (R 2 ) by the rule ( 0,() )(, ) = ()ℎ −1 ().Clearly, the adjoint operator  * 0,() :  2 (R 2 ) →  2 (R + ) is given by The previous operators satisfy the following relations: On the other hand, we introduce the operator  () =  * 0,()  from  2 (C,  || 2 ) onto  2 (R), and its restriction to  2 () (C) is an isometric isomorphism.Thus, the adjoint operator  * () =  *  0,() is an isometric isomorphism from  2 (R) onto the subspace  2 () (C).Hence, these operators satisfy the following relations: Similarly, introduce the isometric embedding  0, : ( 2 (R))  →  2 (R) ⊗  2 (R) by the rule where  = ( 1 , . . .,   ) and and the superscript  means that we are taking the transpose matrix.

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 () = () be a function in  ∞ (R) depending only on  = Re  and we called this function a horizontal symbol.

Definition 1.
Let  be a function in  ∞ (R).The Toeplitz operator with symbol  acting on true-poly-Fock space (or poly-Fock space) is defined as or where  () and   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   with horizontal symbol () ∈  ∞ (R) acting on  2 (C) is unitary equivalent to the multiplication operator   () =  0    * 0 acting on  2 (R), where  0 is defined in Section 2. The function   is given by The following theorem is a generalization of above result for Toeplitz operators with horizontal symbols acting on truepoly-Fock space.
Theorem 2. For any () ∈  ∞ (R), the Toeplitz operator  (), acting on  2 () (C) is unitary equivalent to the multiplication operator  (),  =  ()  (),  * () acting on  2 (R), where the function  (), is given by Proof.We know that the operator  () is unitary and using (20), we obtain that the Toeplitz operator  (), is unitary equivalent to the following operators: Now calculate the explicit expression of the above operator where  ∈  2 (R) and ℎ −1 is the Hermite function of degree  − 1.
We called  (), () the  spectral function for the Toeplitz operator with vertical symbol  in the true-poly-Fock space.
The following result is an extension of above theorem for Toeplitz operators with horizontal symbols acting on poly-Fock space.Theorem 4. For any () ∈  ∞ (R), the Toeplitz operator  , acting on  2  (C) is unitary equivalent to the matrix multiplication operator  , () =    ,  *  acting on ( 2 (R))  , where the matrix-valued function  , = ( ,  ) is given by That is, for ,  = 1, . . ., .
Proof.We have that the operator   is unitary and using (25), we obtain that the Toeplitz operator  , is unitary equivalent to the following operators: Now calculate the explicit expression of the above operator where  = ( 1 , . . .,   ) ∈ ( 2 (R))  and   () is given by ( 22).Therefore we obtain that each component of  , is given by (33), which proves the theorem.
In this section we study the  * -algebra generated by all the Toeplitz operators on  2  (C) with extended horizontal symbols.Definition 6.We define some  * -algebras that we will use in this paper.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.(39) Then the matrix-valued function  , () satisfies Proof.First we consider the case when  − = 0 (analogously we have  + = 0).We proceed to show that the limit value at −∞ of each entry of  , is equal to zero.We know that Let  > 0 be a fixed number; hence there exists Moreover, there exist  > 0 such that |()| <  for  < −.
Under the above assumptions, we estimate the value of each entry of  , () as follows: By similar argument, we obtain lim →+∞  ,  () =  +   .This completes the proof.
Recall that  0 (R) is the set of continuous functions that vanishes at infinity and (R) is the set of continuous functions belonging to  {−∞,+∞} ∞ (R).
The noncommutative Stone-Weierstrass conjecture: let B be a  * -subalgebra of a  * -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  * -algebra type I was proved by I. Kaplansky in [8].In consequence, we have proved that the algebra G   is equal to D. From Corollary 8 we have that the algebra of Toeplitz operators T  −∞,∞ is isometric and isomorphic to algebra D. In summary, we have the following result.