Toeplitz algebras, subnormal tuples and rigidity on reproducing \(C[z_{1},\dots ,z_{d}]\)-modules
It is the aim of the paper under review to study Toeplitz algebras, subnormal tuples, rigidity on reproducing modules \(H\) on a domain \(\Omega\subset{\mathbb C}^d\) over the polynomial ring of \(d\)-complex variables \({\mathbb C}[z_1,\dots,z_d]\) (or on their submodules) in connection to Arveson's boundary representation theory. Several examples prove that the theory is consistent: the Dirichlet module and the Arveson module have the boundary property, while the Hardy module and the weighted Bergman module do not have this property (all modules considered over the unit ball \({\mathbb B}_d)\). If \(H\) has the boundary property, it is proved that there are no inner multipliers for this module (except for constants), hence such a module must be rigid. The same conclusion holds true if \(H\) is given by a positive measure on \({\mathbb C}^d\) (hence \(H\) is subnormal), supported on a set of positive Lebesgue measure. However, a subnormal module \(H\) on a bounded domain \(\Omega\) cannot have the boundary property if \(\bar\Omega\) covers the Taylor spectrum of the tuple \(\{M_{z_1},\dots,M_{z_d}\}\) of coordinate multipliers on \(H\). A \(\mathcal U\)-invariant kernel on the ball \({\mathbb B}^r_d\) (of radius \(r\)) is given by a unique power series \(f\) with nonnegative coefficients and convergence radius \(R\geq r^2\). If \(f\) is regular, then the corresponding reproducing function space \(H^f_d\) is actually a reproducing \({\mathbb C}[z_1,\dots,z_d]\)-module on \({\mathbb B}_d^{\sqrt{R}}\). The Taylor spectrum and the essential Taylor spectrum of \(\{M_{z_1},\dots,M_{z_d}\}\) are computed. An exact sequence of \(C^*\)-algebras is then associated with the Toeplitz algebra \({\mathcal T}^f_d\) on \(H^f_d\). As a consequence of the above results, a necessary condition for modules being subnormal is obtained. Finally, similar properties are studied for submodules \(M\) of Arveson's module \(H^2_d\). It is shown that there are no inner multipliers for \(M\) and that the Toeplitz algebra \({\mathcal T}_d^M\) defined on \(M\) is irreducible and contains all compact operators. The paper is concluded by the remark that two unitarily equivalent submodules of \(H^2_d\) generated by ideals of polynomials must be equal.
scientific article; zbMATH DE number 2091745
scientific article