Convolution operators on standard CR-manifolds. II: Algebras of convolution operators on the Heisenberg group
This paper continues the investigation of the author and \textit{R. Trujillo} in Part I of this paper [ibid. 19, No. 1, 65-104 (1994; see the preceding review)]. The author studies convolution operator algebras with homogeneous kernels on the Hilbert space \(L_2(H_n)\), where \(H_n\) is a Heisenberg group \(H_n = \mathbb{R} \times \mathbb{C}^n\) with the group operation \((t_1, z_1) (t_2, z_2) = (t_1 + t_2 + 2 \text{Im} (z_1, \overline {z}_2), z_1 + z_2)\). The main result of Section 2 is the general local principle for \(C^*\)-algebras based on the Dauns-Hofmann construction. The main results are contained in Sections 3 and 4. There are considered some convolution operator algebras and an isomorphic description of these algebras is given. The author describes spectra and all irreducible representations. In Section 4 a description is given of an algebra generated by Toeplitz operators associated with this Siegel domain.
general local principle for \(C^*\)- algebras
scientific article; zbMATH DE number 638264