C*-algebra generated by horizontal Toeplitz operators on the Fock space

Abstract

We introduce the so-called horizontal Toeplitz operators acting on the Fock space and give an explicit description of the C*-algebra generated by them. We show that any Toeplitz operator with $$L_{\infty }$$ -symbol, which is invariant under imaginary translations, is unitarily equivalent to the multiplication operator by its “spectral function”. This result is also true for the Toeplitz operators whose defining symbols are invariant under translations over any Lagrangian plane. The main result of the paper states 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 $$\mathbb {R}^{n}$$ .