AI 中文总结
该研究针对1<p<∞且p≠2的非交换L^p空间，建立正压缩投影值域完全等距于另一非交换L^p空间时必为完全正投影的结论，还刻画了可压缩分解投影的值域特征。
AI 中文摘要
我们研究非交换L^p空间（其中1<p<∞且p≠2）的子空间的完全等距类与其压缩补空间性质之间的关系。证明：若P∶L^p()→L^p()是正压缩投影，且其值域完全等距于另一非交换L^p空间，则P必为完全正投影。这为文献[ArR24]的主要结果提供了逆命题，且P的正性假设是必要的。我们进一步建立该结果的矩形类似情形，即证明：非交换L^p空间中每个完全等距于形如eL^p(𝒩)(1−e)的矩形非交换L^p空间的闭子空间，均为可压缩分解投影的值域。结合已知的逆蕴含关系，这将可压缩分解投影的值域刻画为恰好是与算子W*三元环相关的矩形L^p空间完全等距的子空间。
英文摘要
We investigate the relation between the complete isometry class of subspaces of noncommutative $\mathrm{L}^p$-spaces and their contractive complementability, where $1 < p < \infty$ with $p \not= 2$. We show that if $P \colon \mathrm{L}^p(\mathcal{M}) \to \mathrm{L}^p(\mathcal{M})$ is a positive contractive projection whose range is completely isometric to another noncommutative $\mathrm{L}^p$-space, then $P$ is necessarily completely positive. This provides a converse to the main result of [ArR24] and the positivity assumption on $P$ is essential. We further establish a rectangular analogue of this result. More precisely, we prove that every closed subspace of a noncommutative $\mathrm{L}^p$-space which is completely isometric to a rectangular noncommutative $\mathrm{L}^p$-space of the form $e\mathrm{L}^p(\mathcal{N})(1-e)$ is the range of a contractively decomposable projection. Combined with the known converse implication, this yields a characterization of the ranges of contractively decomposable projections as precisely the subspaces completely isometric to rectangular $\mathrm{L}^p$-spaces associated with $\mathrm{W}^*$-ternary rings of operators.
Comments22 pages
