Toeplitz operators and Toeplitz algebra with symbols of vanishing oscillation

We study the C*-algebra generated by Toeplitz operators with symbols of vanishing (mean) oscillation on the Bergman space of the unit ball. We show that the index calculation for Fredholm operators in this C*-algebra can be easily and completely reduced to the classic case of Toeplitz operators with symbols that are continuous on the closed unit ball. Moreover, in addition to a number of other properties, we show that this C*-algebra has uncountably many Fredholm components.