AI 中文总结
本文提出基于 Levi 形式的模理论，刻画 Banach 空间压缩投影，并应用于非交换 L^p 空间，证明 2-压缩投影等价于完全压缩，且值域具有三元结构。
AI 中文摘要
我们基于多重次调和函数的 Levi 形式，发展了一种关于复 Banach 空间上压缩投影的微分几何方法。我们证明，若一个投影 $P$ 压缩一个 $\mathrm{C}^2$ 多重次调和函数 $\Phi$，则在其值域的任意一点处，$\Phi$ 的 Levi 形式定义了一个切 Hilbert 几何，使得 $P$ 成为正交投影。当值域为有限维且相关 Levi 形式非退化时，比较与不同点相关的 Levi 几何自然导致正相对算子及相伴的单参数群。我们将此框架应用于迹非交换 $\mathrm{L}^4$-空间。利用对偶映射的三次形式，我们以 Jordan 三元正交性刻画压缩投影，并证明 $\mathrm{L}^4(\mathcal{M})$ 上的投影是 $2$-压缩的当且仅当它是完全压缩的。当环境 von Neumann 代数为有限维时，我们将值域上左、右切形式相关的相对算子识别为链接 von Neumann 代数上模算子平方根逆的限制。这意味着任何 $2$-压缩投影的值域完全等距于与 $\mathrm{W}^*$-三元算子环相关的矩形非交换 $\mathrm{L}^4$-空间。因此，每个这样的投影都是压缩可分解的。最后，对于每个 $1 < p < \infty$ 且 $p \neq 2$，我们证明迹非交换 $\mathrm{L}^p$-空间上的每个有限秩 $2$-压缩投影都是完全压缩的。对于 $p > 2$，我们获得其值域关于有限维 $\mathrm{W}^*$-$\mathrm{TRO}$ 的加权三元描述。
英文摘要
We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\mathrm{C}^2$ plurisubharmonic function $Φ$, then, at any point of its range, the Levi form of $Φ$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\mathrm{L}^4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\mathrm{L}^4(\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\mathrm{L}^4$-space associated with a $\mathrm{W}^*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 < p < \infty$ with $p \neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\mathrm{L}^p$-space is completely contractive. For $p > 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\mathrm{W}^*$-$\mathrm{TRO}$.
Comments74 pages
