Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space of ℂn

We introduce the so-called extended Lagrangian symbols, and we prove that the C∗-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.