Let \({\mathscr {M}}\) be a semi-finite von Neumann algebra with a faithful normal semi-finite trace \(\tau \) and \(L^{p}({\mathscr {M}})\) the non-commutative \(L^p\) space associated with \(({\mathscr {M}},\tau )\). We extend star partial order \(\overset{*}{\le }\) and diamond order \(\le ^{\diamond }\) to \(L^{p}({\mathscr {M}})\) and present some properties about several bounds of these two partial orders. We establish a result about the existence of the star infimum and supremum. We also prove that a subset with an upper bound must have a minimal upper bound in \(L^p({\mathscr {M}})\) under the diamond order in the case of finite von Neumann algebra \({\mathscr {M}}\). However, we give an example and show that this result may fail if \({\mathscr {M}}\) is not finite. Moreover, we characterize the forms of all norm closed hereditary subspaces in \(L^{p}({\mathscr {M}})\) under these two partial orders.

令\({\mathscr {M}}\)是一个半有限冯诺依曼代数，具有忠实的正规半有限迹\(\tau \)和\(L^{p}({\mathscr {M}} )\)与\(({\mathscr {M}},\tau )\)关联的非交换\(L^p\)空间。我们将星形偏序\(\overset{*}{\le }\)和菱形序\(\le ^{\diamond }\)扩展到\(L^{p}({\mathscr {M}})\ )并给出关于这两个偏序的几个边界的一些性质。我们建立了关于恒星下界和上界存在的结果。我们还证明了具有上限的子集在\(L^p({\mathscr {M}})\)在有限冯诺依曼代数\({\mathscr {M}}\)的情况下，在菱形阶次下。然而，我们举了一个例子并表明如果\({\mathscr {M}}\)不是有限的，这个结果可能会失败。此外，我们在这两个偏序下刻画了\(L^{p}({\mathscr {M}})\)中所有范数闭遗传子空间的形式。