A Note on C*-Algebra of Toeplitz Operators with L-Invariant Symbols

Let L c Cn be a Lagrangian plane. In this article, we give structure for L-invariant operators on the Fock space F2(Cn). With the help of this structure, we study Toeplitz operators Ta on F2(Cn) with L-invariant symbols a E Lm(Cn). We show that every operator in the C*-algebra generated by Toeplitz operators with L-invariant symbols, denoted by TL(Lm), can be represented as an integral operator of the form ?(H?X f)(z) = Cnf (w)?(z + X* Xw)ez wd?(w) for some ? E F2(Cn) and X E U(n, C) such that XL = iRn. In fact, we prove that H?X E TL(Lm) if and only if there exists m E Cb,u(Rn) such that ? 2?n/2 ?(z) =p ?Rn m(x)e-2(x-X2z )2dx,z E Cn. Here Cb,u(Rn) denotes all functions on Rn which are bounded uniformly continuous with respect to the standard metric on Rn.