Radial Toeplitz Operators Revisited: Discretization of the Vertical Case

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 VSO ( 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 VSO ( R + ) of “very slowly oscillating functions”, i.e. the bounded functions that are uniformly continuous with respect to the logarithmic distance ρ ( 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.