  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 sharper estimates of
noncommutative Kakeya maximal functions and geometric estimates in the plain.
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. We point out that even
geometric estimates in the plain are different from that in the commutative
case.
