We prove that spectral projections of Laplace-Beltrami operator on the $m$-complex unit sphere $E_{\Delta_{S^{2m-1}}}([0,R))$ are uniformly bounded as an operator from $H^p(S^{2m-1})$ to $L^p(S^{2m-1})$ for all $p\in (1,\infty)$. We also show that the Bochner-Riesz conjecture is true when restricted to cylindrically symmetric functions on $\R^{n-1} \times \R$.