The K-theory of classical complex projective spaces was computed first time by Atiyah and Todd
with use of tools which are not available in noncommutative geometry.
Recently, the K-theory of a quantization of complex projective spaces, defined as a replacement of all discs
in the classical pushout by quantum discs, has been computed by Albert Sheu in terms of Toeplitz projections.
The main difficulty in relating his result with the classical result of Atiyah and Todd is that the Toeplitz projections
do not admit a classical limit. We overcome this difficulty in the case of a complex projective plane,
reobtaining the result of Atiyah and Todd in a way admitting a quantization, and an explicit homotopy
between a quantized Milnor type idempotent, coming from a clutching construction, and the Toeplitz type projection of Sheu.