  In this paper, we establish the full $L_p$ boundedness of noncommutative
Bochner-Riesz means on two-dimensional quantum tori, which completely resolves
an open problem raised in \cite{CXY13} in the sense of the $L_p$ convergence
for two dimensions. The main ingredients are sharp estimates of noncommutative
Kakeya maximal functions and geometric estimates in the plane. We make the most
of noncommutative theories of maximal/square functions, together with
microlocal decompositions in both proofs of sharper estimates of Kakeya maximal
functions and Bochner-Riesz means.
