Radial Toeplitz Operators Revisited: Discretization of the Vertical Case

Abstract

It is known that radial Toeplitz operators acting on a weighted Bergman space of the analytic functions on the unit ball generate a commutative C*-algebra. This algebra has been explicitly described via its identification with the C*-algebra \({{\rm VSO}(\mathbb{N})}\) of bounded very slowly oscillating sequences (these sequences was used by R. Schmidt and other authors in Tauberian theory). On the other hand, it was recently proved that the C*-algebra generated by Toeplitz operators with bounded measurable vertical symbols is unitarily isomorphic to the C*-algebra \({{\rm VSO}(\mathbb{R}_+)}\) of “very slowly oscillating functions”, i.e. the bounded functions that are uniformly continuous with respect to the logarithmic distance \({\rho(x,y)=|\ln(x)-\ln(y)|}\). In this note we show that the results for the radial case can be easily deduced from the results for the vertical one.