m -Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the m -Berezin transform as a linear operator from the space of bounded operators to L^{\infty} is found. We show that the m -Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the m -Berezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.